APPROXIMATIONS TO THE NONCENTRAL CHI-SQUARE

AND NONCENTRAL F DISTRIBUTIONS

by

BILL RANDALL WESTON, B.A.

A THESIS

IN

MATHEMATICS

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

>1ASTER OF SCIENCE

Approved

Accepted

December, 197 3

Cep . ^

ACKNOWLEDGEMENTS

I am deeply indebted to Professor James M. Davenport

for his direction of this thesis and to other members of

my committee. Professor Benjamin S. Duran and Professor

Truman 0. Lewis, for their assistance.

I would also like to thank my wife Sue for her help

and support during the preparation of this paper.

11

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ^^

LIST OF TABLES ^

I. NOTATION 1

II. INTRODUCTION 3

I I I . TWO AND THREE MO!ffiNT APPROXII'lATIONS TO x / . ^ • "7 ' (n,A)

Introduction 7 Patnaik's Two Moment Central x Approximation. 8 Pearson's Three Moment Central x Approxi­mation 8 Specific Approach to the Problem of Percentage

Point Generation 9

General Results of the Approximations 10

Results at the 10% Level 11

Results at the 5% Level 11

Results at the 2.5% Level 12

Summary 13

IV. TWO AND THREE MOMENT APPROXIMATIONS TO

F' , . , 18 (n,d,A)

Introduction 18

Patnaik's Two Moment Central F Approximation . 19

Tiku's Three Moment Central F Approximation. . 19

Chaubey's Approximation 20 Specific Approach to the Generation of Approx­imation Points 21

General Results 22

A Warning Note 23

Results at the 10% Level 2 3

Results at the 5% Level 23

Results at the 2.5% Level 24

Summary 25

111

Page

V. FOUR MOMENT APPROXIMATIONS TO THE NONCENTRAL

DISTRIBUTIONS 34

Introduction 34

Indices of Skewness and Kurtosis and Moments Needed in the Four Moment Approach 34

General Approach to the Four Moment Method . . 35

Approach to the Four Moment Method Using CORFIS 36

Summary of Results Using the CORFIS Four Moment Approach 39

Noncentral F Results 39 2

Noncentral x Results 42

10% Level - X/^ x 42 ' (n,A)

5% Level - X/^ A 43 ' (n,A)

2.5% Level - X/^ AX 45

^n, A;

General Summary of the Four Moment Approach. . 46

A Final Comment on the Four Moment Method. . . 47

VI. AN EXAMPLE OF THE USE OF CORFIS OBTAINED NON-

CENTRAL POINTS IN A MODIFIED SATTERTHWAITE PRO­

CEDURE 49

Introduction 49

The Usual Satterthwaite Technique 49

The Modified Satterthwaite Technique 50 Specific Application of the Modified Satterth­waite Approach 52 Summary 54

APPENDIX

SELECTED APPROXIMATE PERCENTAGE POINTS OF

X\ ^j AND F' ^ ^ j USING CORFIS METHODS . . . 55

REFERENCES ^2

IV

LIST OF TABLES

TABLE PAGE 2

1. Analysis of Patnc.ik's Approximation to x\ A ) * • -15 2

2. Analysis of Pearson's Approximation to X/ .x . . . 16 ^n. A;

3. Example of Mixed Increasing and Decreasing Behavior Pearson Approximation - Lower 10% Points 17

4. Analysis of Error for Approximations to the Upper 10% Points of F' -, ,, 27

(n,d,A) 5. Analysis of Error for Approximations to the Lower

10% Points of F' . ,, 28 (n,d,A)

6. Analysis of Error for Approximations to the Upper 5% Points of F' , ,, 29 - (n,d,A)

7. Analysis of Error for Approximations to the Lower 5% Points of F', , ,, 30 - (n,d,A)

8. Analysis of Error for Approximations to the Upper 2.5% Points of F' . ,, 31

(n,d,A) 9. Analysis of Error for Approximations to the Lower

2.5% Points of F', , ,, 32 (n,a,A)

10. Analysis of Error for the Patnaik Approximation to 10% Points of F' , ,v 33

(n,d,A) 11. Four Moment .vs. Three Moment For Upper 10% -

Y'2 43 ^(n,A)

12. Percentage Errors for the Four and Three Moment Approximations to xj^ A) ^^ ^^^ Lower 10% Level. . 44

13. Four Moment .vs. Three Moment for Upper 5% -Y.2 45 ^(n,A)

14. Comparisons of the Four and Three Moment Fits to X/^ AX in the Upper Tail 45 ^(n,A)

15. Selected Approximate Upper 10% Points of the Non-Central Chi-Square using the Three Moment Pearson Approximation 56

16. Selected Approximate Lower 10% Points of the Non-Central Chi-Square using the Three Moment Pearson Approximation 57

17. Selected Approximate Upper 10% Points of the Non-Central Chi-Square using the Two Moment Patnaik Approximation ^^

V

Table PAGE

18. Selected Approximate Lower 10% Points of the Non-Central Chi-Square Using the Two Moment Patnaik Approximation 59

19. Selected Approximate Upper 2.5% Points of the Noncentral Chi-Square Using the Three Moment Pearson Approximation 60

20. Selected Approximate Lower. 2.5% Points of the Non-Central Chi-Square Using the Three Moment Pearson Approximation 61

n,d,A) • . • .62

n,d,A) . . . .63

n,d,A) . . . .64

n,d,A) . . . .65

n,d,A) . . . .66

n,d,A)

21. Selected Approximate Lower 10% Points of F

Using the Patnaik Two Moment Approximation

22. Selected Approximate Lower 10% Points of F

Using the Chaubey Two Moment Approximation

23. Selected Approximate Lower 10% Points of F

Using the Tiku Three Moment Approximation.

24. Selected Approximate Upper 10% Points of F

Using the Patnaik Two Moment Approximation

25. Selected Approximate Upper 10% Points of F

Using the Chaubey Two Moment Approximation

26. Selected Approximate Upper 10% Points of F

Using the Tiku Three Moment Approximation 67

27. Selected Approximate Lower 5% Points of F', ^ .

Using the Chaubey Two Moment Approximation . . . .68

28. Selected Approximate Lower 5% Points of F', ^ ^.

Using the Tiku Three Moment Approximation 69

29. Selected Approximate Upper 2.5% Points of FJ^^^^^^

Using the Tiku Three Moment Approximation 70

30. Selected Approximate Upper 2.5% Points of FJ^ ^

Using the Chaubey Two Moment Approximation . . . .71

31. Selected Approximate Lower 10% Pts. of F' ^ ^ j

Using the Four Moment Approach; A = 6.0 72

32. Selected Approximate Upper 10% Pts. of F' ^ ^ ^

Using the Four Moment Approach; A = 2.0 73

33. Selected Approximate Lower 5% Pts. of F' ^ ^ ^

Using the Four Moment Approach; A = 14 74

VI

Table Page

34. Selected Approximate Upper 2.5% Pts. of F', j AN

Using the Four Moment Approach; A = 20 75

35. Selected Approximate Upper 10% Points of the Non-Central Chi-Square Using the Four Moment Method . .76

36. Selected Approximate Lower 10% Points of the Non-Central Chi-Square Using the Four Moment Method . .77

37. Selected Approximate Upper 5% Points of the Non-Central Chi-Square Using the Four Moment Method . .78

38. Selected Approximate Lower 5% Points of the Non-Central Chi-Square Using the Four Moment Method . .79

39. Selected Approximate Upper 2.5% Points of the Non-Central Chi-Square Using the Four Moment Method . .80

40. Selected Approximate Lower 2.5% Points of the Non-Central Chi-Square Using the Four Moment Method . .81

Vll

CHAPTER I

NOTATION

The items listed below are used repeatedly in the

sequel. Most items are defined and explained as they

appear in the text; but in order to give the reader a

summarized listing of the most frequently used notation

and conventions, the following section has been prepared.

R.V. - random variable

d.f. - degrees of freedom

n or n.d.f. - numerator degrees of freedom

d or d.d.f. - denominator degrees of freedom

A - noncentrality parameter associated with the

noncentral Chi-Square R.V.

or the noncentral F R.V.

U - mean of the R.V. X; i.e., E[x] = y' = p

u - rth central moment of the R.V. X; i.e., E[(X-y) ] = y r ^

2 3 3, - an index of skewness. (3-j = y3/y2-)

2 32 - an index of kurtosis. (32 = y4/y2)

X=Y - The R.V. X is approximately distributed as the R.V.Y.

.Ox;.x - "x" is used in the text and tables to indicate

that the digit is variable from 0 to 9, unless

specified otherwise.

3(p,q) - central Beta distribution with parameters p and q.

X^ - central Chi-Square distribution

F - Snedecor's central F distribution

F'(n,d,A) - noncentral F distribution with n and d d.f.

and noncentrality parameter A. The noncentral

F distribution will be defined throughout as

in Johnson and Kotz (1970). Also, the distribu­

tion will be cited in the text as "the noncen­

tral F."

.2

(n A) ~ '^o^central Chi-Square distribution with d.f. n and

noncentrality parameter A. The noncentral Chi-

Square is defined in this paper as in Johnson and

Kotz (1970). This distribution will be referred 2

to as "the noncentral x •"

absolute error - |approximate percentage point - exact

percentage point|. In the computation of

absolute error, the approximate point ob­

tained from the approximation was rounded

to the same number of significant digits

as was in the exact point before any sub­

traction was performed. Also, most exact

points used have been rounded to a certain

number of significant digits. Thus, in

absolute error considerations the round­

ing procedures must be kept in mind. percentage error - (absolute error/exact point) x 100

CHAPTER II

INTRODUCTION

2

Numerous approximations to the noncentral x and non-

central F distributions have been proposed and compared in

the literature. The bulk of the comparisons and studies

of these approximations are concerned with evaluation of

the probability integral of the noncentral F and noncentral 2

X . Not many comparisons of the approximations have been

made from the standpoint of percentage points of the non-

central distributions. Percentage points are useful in

Monte Carlo studies and in tests of hypotheses that involve 2

either the noncentral F or noncentral x distributions.

To meet the demand for percentage points various approxi­

mations have been developed. Exact methods involving

iterative techniques are usually not employed to obtain

"exact" points because convergence may be very slow.

Thus, since these exact percentage points may be difficult

to obtain simple and reasonably accurate approximations

2

to the noncentral x and F are used. After examining the

literature, the simplest yet sufficiently accurate approxi­

mations were determined. The three best (in terms of accur-2

acy and simplicity) approximations to the noncentral x are 2

the Patnaik (1949) two moment central x ; the Pearson 2

(1959) three moment central x /* and the Johnson and Pearson

(1969) four moment fit. The simplest, yet very accurate,

approximations to the noncentral F were discovered to be

the Patnaik (1949) two moment central F; the Tiku (1965)

three moment central F; the Mudholkar et al. (1976) two

moment central F; and a four moment method based on a

Pearson type VI distribution.

One of the purposes of this thesis is to evaluate the

accuracy of the above approximations with respect to the

generation of percentage points. It was found that a

great deal of comparison has not been carried out with re-

2

gard to percentage points of the noncentral x and noncen­

tral F distributions. The major stumbling block to achiev­

ing comparisons with exact points is that fractional d.f. 2

are involved in the central x and central F distributions

in the two and three moment approximations to the noncen­

tral distributions. That is to say, much labor and time

are involved in interpolating in the existing central dis­

tribution tables of percentage points. In the case of the

four moment methods, interpolation must be carried out in

tables provided by Pearson and Hartley (1972), Johnson et

al. (1963), or Bower et al. (1974) . Therefore, to over­

come the problems of interpolation and the limitations

placed on the number of points that can be obtained be­

cause the existing tables have a finite parameter range,

an efficient and expedient method of obtaining approxi­

mate percentage points has been developed. Development of

a method that allows one to obtain a wide range of approxi­

mate percentage points quickly and easily is a second

major objective achieved in this thesis.

Essentially, the method consists of using the previously

mentioned approximations in conjunction with a. Fortran sub­

routine written by Herring and Davenport (Herring, 1974).

The subroutine CORFIS is employed to obtain central F, 2

central x / and central Beta (Pearson type I) percentage

points. CORFIS makes use of a modified Cornish Fisher

expansion to produce percentage points at the upper and

lower 10%, 5%, and 2.5% levels for the central F, central

X, central Beta, and other Pearson curves (except Type IV).

Major input parameters to CORFIS are the numerator d.f.

and denominator d.f. for a central F distribution. The

subroutine is valid for real-valued degrees of freedom

greater than or equal to 1. Other input parameters are

used to select the desired tail of the F distribution and

to specify the probability level desired. The output

variable of CORFIS is the desired percentage point of the

central F distribution. Percentage points of other Pearson

curves can be obtained as explained in Herring (1974). This

computing algorithm will compute Pearson type VI (essentially

the F distribution) percentage points accurate to +1 in 3

digits in approximately 11 milliseconds per subroutine

call using double precision on an IBM 370/145.

A final objective of this paper is to apply some of

the moment fitting approximations previously mentioned in

a modified Satterthwaite (1946) procedure. The modification

of the Satterthwaite method involves a procedure in which

the numerator of the Satterthwaite F is assumed to be a

2 2 noncentral x instead of a central x • Results of this effort will be presented in the concluding chapter.

CHAPTER III

TWO AND THREE MOMENT APPROXIMATIONS TO X'^ (n,A)

Introduction

Of the many approximations to the noncentral Chi-Square

the simplest yet quite accurate are Patnaik's (1949) two

moment central Chi-Square, Pearson's (1959) three moment

central Chi-Square, and a four moment Pearson type I (Beta

distribution) approach (Johnson and Pearson, 1969). The

four moment approach will be examined in a later chapter.

Other prominent noncentral Chi-Square approximations in­

clude Tiku's (1965) Laguerre series approximation and

several approximations involving a normal transformation

proposed by various authors including Abdel Aty (1954)

and Sankaran (1963). The Tiku approximation is not examined

because the series approximation is very complex; the calcu­

lation of large cumulants is difficult; and the inversion

of the series required to obtain percentage points is an

impediment. The various normal approximations are fairly

simple to use to obtain percentage points, but they have

been shown to be no more accurate if not as accurate as

the Pearson approximation (Johnson and Kotz, 1970). Thus,

for the above reasons, approximations other than the two

and three moment ones are not explored in the sequel. One

might also consider an Edgeworth expansion to improve any

of the above approximations. However, the Edgeworth improve-

8

ment is not considered because the need to compute high

order cumulants for the noncentral x makes it an unattrac­

tive alternative.

Patnaik's Two Moment Central x Approximation

The Patnaik approximation may be expressed as follows: 2

Suppose X represents a noncentral x with n d.f. and 2

noncentrality parameter A(X~x'/ -v \ ) • Also, let Y represent ^n, A;

2 2 a central x R.V. with v d.f. (Y-X/ \)« The Patnaik approxi mation is obtained by replacing X by a multiple of a central 2

X / say pY. Let the fact that X is approximately distribu­

ted as pY be denoted by X = pY. If p and v are chosen so

that the first two moments of X and pY agree one obtains p = (n+2A)/(n+A) and v= (n+A)^/(n+2A). (2.1)

Thus, the exact percentage point x of X is approximated

using x = py , where y is the a probability point of a

central Chi-Square Y with v d.f.

2 Pearson's Three Moment Central x Approximation

Suppose X and Y are defined as above. Additionally,

assume X = pY + b, where p, v, and b are constants to be

obtained. If one equates the first three moments of X

and pY + b and solves for p, v, and b, one attains the

following:

p = (n+3A)/(n+2A)

V = (n+2A)^/(n+3A)^ (2.2)

b = -AV(n+3A) .

The exact percentage point x of X is approximated accord-

ing to X = py + b, where y is the a probability point

of a central Chi-Square Y with v d.f.

Specific Approach to the Problem of Percentage Point Gen­

eration

At the outset it was desired to verify the accuracy

of both the two and three moment fits for a large range of

2 the parameters A and n of X/ -v \ • The only comparisons found

in, A;

in the literature are at the upper and lower 5% level for

n = 2, 4, 7 and A = 1, 4, 16, 25 (several sources includ­

ing Patnaik, 1949 and Johnson and Kotz, 1970). Thus, in

order to establish the accuracy of the moment fits for a

greater number of points, approximate percentage points at

the upper and lower 10%, 5%, and 2.5% levels were obtained

for n = 1(1) 12, 15, 20 and /A = .2(.2)6. These percen­

tage points were obtained quickly and without interpolation

problems with the help of CORFIS (Herring, 1974). The

obtained approximate points were compared to exact points

provided by Johnson (1968). Johnson's percentage points

are correct to +1 in the the fourth significant digit. In

the comparisons of exact points with approximate points,

10

the approximate points were rounded so that 4 significant

digits are retained in the point. This rounding procedure

was carried out in all cases before absolute error was com­

puted. Extensive analyses of the two approximations were

carried out to establish the accuracy of both the two and

three moment approximations as A or n of xJ AX varies.

These analyses are presented in tables 1, 2, and 3. The

provided tables should help one judge the accuracy of the

approximations for specific n and A and will provide one

with a general yardstick to measure the accuracy of the

four moment approach presented in Chapter V.

General Results of the Approximations

In the explanations of the results that follow, refer­

ence is made to absolute error and percentage error. These

terms are defined as

absolute error = |approximate point - exact point

and

percentage error = (absolute error/exact point) x 100.

Analyses of the approximations are given in tables 1,

2, and 3, with accuracy trends being analyzed with respect

to absolute error. Also, selected approximate percentage

points are presented in tables 15-20 in the appendix.

For all percentage levels examined one general result

holds; namely, the accuracy of both approximations increases

11

2 as n of X/j \ increases.

Results at the 10% Level

For the upper 10% points examined, the Patnaik approxi­

mation was determined to be as accurate if not more accurate

than the Pearson fit. The Pearson approximation is more

accurate than Patnaik's at A = 33.64 and 36. Secondly,

on the basis of the data explored, it seems as if there

exists a slight trend for Pearson's approximation becoming 2

more accurate than Patnaik's as A of xV AX increases. ' (n,A)

Finally if one examines tables 1 and 2, it is recognized

that the Patnaik approximation becomes less accurate as

A increases, whereas the Pearson approximation tends to­

ward a general error pattern for a A range as n varies

from 1 to 20.

At the lower points, the Pearson approximation is the

more accurate method. One must be aware, though, that for

small n(n = 1 or 2) with small A (say A_<4) the Pearson

approximation yields negative percentage points. For the

large A; e.g. AM, the Pearson fit is significantly better

than Patnaik's method. The two methods appear to be very

comparable for small A and small n. Other accuracy con­

siderations can be gained by referring to tables 1 and 2.

Results at the 5% Level

After considering Patnaik's and Pearson's approxima-2

tions at the upper 5% points of the noncentral x / it was

12

discovered that the Pearson approximation is more accurate

for AM. One can see this rather clearly as the Pearson

approximation is increasing in accuracy as A increases

(i.e., for A>_4) , and the Patnaik fit is decreasing in

accuracy as A increases. For A0.24, the Patnaik approxi­

mation is the more accurate method to obtain upper 5% points.

Except where the Pearson approximation gives negative

points (primarily for small n = 1 or 2 with A£4), the

Pearson approximation is substantially better than Patnaik's

procedure in generating lower 5% points. The superiority

of Pearson's fit is especially evident for larger A. For

example, at A = 12.96 the Pearson absolute error ranges

from .03 to .10, whereas the Patnaik error ranges from .61

to .14.

Results at the 2.5% Level

For most of the upper 2.5% points; that is, for Ay. 44,

the three moment fit is more accurate than Patnaik's. The

Pearson absolute error for A> 1.44 is .01 or 0 versus increas­

ing Patnaik absolute error as A increases. In the range of

A£l, Pearson's approximation and Patnaik's fit are nearly

equivalent in terms of accuracy for the upper tail of

x'2 ^(n,A)-

In the lower tail, the Patnaik approach does not miss

the exact point as badly as does Pearson's for small A

with n = 1. Mainly, this result comes from the negative

13

point generation of the Pearson fit. After considering the

approximations at other combinations of A and n, it is evident

that Pearson's method is more accurate than Patnaik's.

Summary

In general, the three moment Pearson approximation is

more accurate than the Patnaik two moment fit. However,

one must beware that the Pearson method will give negative

values for lower percentage points for some combinations

of small A with small n. The results of the two approxi­

mations' accuracy have been tabled and should provide a

basis to judge the adequacy of the methods in generating

percentage points. CORFIS, when utilized with the approxi­

mations, gives one an expedient and efficient procedure to

obtain approximate percentage points for a wide range of 2

the parameters of x} ^x•

14

EXPLANATIONS PERTAINING TO TABLES 1 AND 2

The following tables 1 and 2 are concerned with the

accuracy of the Patnaik and Pearson approximations to

X'^

(1) n = 1(1)12, 15, 20

/I = .2(.2)6

(2) The general trend shown in the tables is obtained

using the criterion of absolute error.

(3) absolute error = |approximate percentage point-exact

point I.percentage error = (absolute error/exact point)

X 100.

(4) Entries within parentheses are percentage errors.

(5) When "x" appears in an entry, this signifies that the

digit can vary from 0 to 9. In other words, the fig­

ure is given merely to portray an approximate magni­

tude.

(6) Average absolute error is obtained for each A range.

Each A entails 14 error values (one error for each n,

as n = 1(1)12, 15, 20) .

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17

Table 3

EXAMPLE OF MIXED INCREASING AND DECREASING BEHAVIOR

PEARSON APPROXIMATION - LOWER 10% POINTS

^ 1

2

3

4

5

6

7

8

9

10

11

12

15

20

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2.56

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.0000

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.0010

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.0020

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.0000

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3.24

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4.00

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.0000

*Tabled values are absolute errors (|approx. pt. - exact

pt.I).

Mixed increasing and decreasing behavior means that as A in­

creases the approximation becomes more accurate for the 2

smaller n of x ^^^ less accurate for the larger n. (n,A)

The magnitude of n for this trend to exist varies with the

size of A. That is to say; the larger A is, the larger

n can be for the trend to exist.

CHAPTER IV

TWO AND THREE MOMENT APPROXIMATIONS TO F', ^ , (n,d,A)

Introduction

In this chapter, three approximations are explored

with respect to the generation of percentage points of the

noncentral F distribution. A fourth approximation, a four

moment method, is examined in Chapter v. The simple yet

sufficiently accurate approximations analyzed in this sec­

tion are Patnaik's (1949) two moment central F; Tiku's

(1965) three moment central F; and the Mudholkar, Chaubey,

and Lin (1976) approximation (henceforth to be referred to

as the Chaubey approximation). Other well known approxima­

tions which are not investigated are Tiku's (1965) Laguerre

series approximations; Severe and Zelen's (1960) normal

approximations; and the Mudholkar et al. (1976) Edgeworth

approach. Tiku's Laguerre series approximations are rather

complex expressions which are difficult to invert. Further­

more, Tiku (1966) noted that the series approximations are

not as accurate as his three moment approach. Severe and

Zelen's approximation has merit in terms of simplicity, but

it is not examined because it has been shown to be no more

accurate than Tiku's or Chaubey's approximations (see Tiku,

1966). Chaubey's Edgeworth approximation qualifies as a

very accurate approximation, but the inversion problem and

the labor of computing large cumulants tend to make it un­

appealing.

19

Since Patnaik's, Tiku's, and Chaubey's approximations

will be explored, the forms of the respective methods are

now presented.

Patnaik's Two Moment Central F Approximation

Suppose F* represents a noncentral F with numerator

d.f. n, denominator d.f. d, and noncentrality parameter

A. (i.e., F*~Fl^ d A)^* Furthermore, let F represent a

central F R.V. with numerator d.f. v and denominator d.f.

d (i.e., F-F,^^ ^ , ) .

The Patnaik approximation results from replacing F*

by a multiple of a central F, say pF. Let the fact that

F* is approximately distributed as pF be denoted by F* = pF.

The constants p and v are Solved for after equating the first

two respective moments of F* and pF. The solution of the two

equations yields

p = (n+A)/n and v = (n+A)^/(n+2A). (3.1)

Finally, the exact percentage point f* of F* is approximated

using the relation f* = pf , where f is the a probability

point of a central F distribution with parameters v and

d.

Tiku's Three Moment Central F Approximation

Let F*~F', . ,v and F~F, , .v. Assume F* = pF -i- b, (n,d,A) (v,d)

where v, p and b are to be obtained. Tiku determined p, v,

and b by equating the first three moments of F* and pF + b.

20

The solutions for p, v, and b are given as;

V = i(d-2)[(E/(E-4))^/2 -1] (3.2)

p = (v/n) [H/((2v + d - 2)K)]

b = [d/(d-2)][(n+A)/n - p]

H = 2(n+A)^ -I- 3(n+A) (n-J-2A) (d-2) + (n+3A)(d-2)^

K = (n+A)^ + (d-2) (n+2A) 2 3 E = H /K" .

The exact percentage point f* of F* is approximated by f* =

pf + b, where f is the corresponding a probability point

of a central F distribution with v and d d.f.

Chaubey's Approximation

Suppose F* and F are defined as above in the Tiku

derivation. Also, as in the Tiku approach it is assumed

F* = pF -I- b. However, in Chaubey's approximation v is set

equal to the d.f. used in the Pearson three moment approxi-

2

mation to the noncentral x (see relations 2.2) . The vari­

ables p and b are found by equating the first two moments

of F* and pF + b. Upon solving the two equations one

obtains:

21

p = (v/n) Cv^ -f (d-2)v)"-^/^ X

[(d-2) (n4-2A) + Cn-t-A) ]- / C3.3)

b = d (d-2)-•'•CI + A/n - p)

3 2 with V set equal to Cn+2A) /(n+3A)

Approximate percentage points are obtained as explained

in the Tiku presentation above.

Specific Approach to the Generation of Approximate Points

Approximate upper and lower 10%, 5 %, and 2.5% points

of F', J -v \ were generated using the previously described ^n,a,Ay

approximations for the following parameter values:

A = 2, 6, 10, 14, 20

n = 1, 2, 3, 5, 10, 15, 30, 60

d = 4, 6, 8, 10, 20, 30, 40, 60.

2

As in the case of the noncentral x point generation, the

subroutine CORFIS (Herring, 1974) was used to obtain non-

integer d.f. central distribution (in this case the central

F distribution) percentage points. The three approxima­

tions were compared to each other in order to ascertain

the most accurate method of generating approximate percent­

age points. Tables provided by Lachenbruch (1966) were

used as the source of exact points in all comparisons.

Values in his tables are generally correct to four decimal

places. Three significant figures are given for n.d.f. = 1;

22

d.d.f. > 30; n.d.f. = 2, 3 with d.d.f. = 4; and n.d.f. = 2

with d.d.f. = 6. The number of significant digits retained

for the approximate points is adjusted to conform to the

particular region of Lachenbruch's tables.

General Results

A primary objective of the examination of the moment

fitting approximations was to discover the effect of the

parameters A, d, and ^ of F', , ,. on the accuracy of each

approximation. In all examinations one parameter was allowed

to vary while the other two were held fixed. The most ob­

vious relationship found was that accuracy at all probabil­

ity levels for all approximations increases as n increases.

Also, all three moment fitting approximations give quite

analogous results when n > 30. Relationships are not clearly

apparent between A and accuracy or d and accuracy. However,

in the case of the Patnaik method for the lower percentage

points it was ascertained that the accuracy of the approxi­

mation decreases with increasing d of FJ . ,v .

In view of the fact that little or no trends with re­

spect to the two parameters A and d were observed, tables

concerning the analysis of accuracy are presented with a

concentration on the effects of n of F|^ ^ v on the accur­

acy of the approximations. It was also discovered that

Tiku's and Chaubey's approximations are more accurate than

23

Patnaik's; thus, the decision was made to present more

tables on the results of the Chaubey and Tiku methods.

Tables 4-9 illustrate the general accuracy of the Chaubey

and Tiku approximations. An example of the Patnaik approxi­

mation accuracy is given in table 10. Tables 21-30 of the

appendix illustrate the accuracy of the three moment fitting

approximations for selected approximate percentage points.

A Warning Note

In interpreting the results that follow, one should be

forewarned that both Tiku's and Chaubey's approximations give

negative percentage points for small n coupled with small

A (e.g., n = 1 with A = 2) in the lower tails of FJ^ ^y

Results at the 10% Level

At the 10% level, for both upper and lower tails, the

most accurate approximation of the three is the Tiku method.

One exception exists at A = 2 for the lower points where

Chaubey's fit is superior. The Tiku and Chaubey methods

are very comparable when n of FJ^ j is greater than or

equal to 10.

Results at the 5% Level

For the upper tail 5% points considered, Tiku's and

Chaubey's approximations yield generally the same accuracy.

Both approximations are superior to Patnaik's, especially

for n < 15.

24

With respect to the generation of lower 5% points the

following emerges:

(1) For A £ 10 with n £ 10 (or 15 in some cases),

Chaubey's method is superior.

(2) For A > 14 with n £ 10 (or 15 for some cases),

Tiku's approximation is superior.

(3) For n >_ 15, Tiku's and Chaubey's approximations

yield generally equivalent results.

(4) Patnaik's fit is inferrior except for n >_ 30

where all three methods are fairly analogous approxi­

mations.

Results at the 2.5% Level

As in the case of the upper 5% level, the Tiku and

Chaubey methods are comparable. Both are more accurate

than Patnaik's fit, especially for n £ 15. There is some

sporadic Chaubey method superiority for A £ 10 in the upper

tail and some isolated Tiku method superiority for A > 14.

In the lower tail, Chaubey's approximation appears to

be the most accurate of the three. The following should

give one a reasonable idea of the approximations' behavior:

(1) For A = 14 with d = 4, 6 as well as A = 20 with

d = 4, 6, 8, 10; there is some Tiku approximation

superiority.

(2) Chaubey's approximation is superior for smaller

n, usually n < 10.

25

(3) For n >_ 30, Patnaik's approximation gives generally

the same results as the other two moment fits.

Summary

For the range of parameters examined, the major trend

discovered is that the accuracy of all moment fitting methods

increases as n of F', , ,. increases. Also, it was observed (n,a,A)

that the three examined approximations become quite similar

for large n.

It is hoped that the tables presented for selected

percentage points along with the overall accuracy analysis

tables will give one an adequate basis to judge the accur­

acy of the discussed approximations. Furthermore, one may

use these analyses to establish some measure of the accur­

acy of the four moment method presented subsequently.

26

EXPLANATIONS PERTAINING TO TABLES 4-10

The following tables show the analysis of error for

the Tiku, Patnaik and Chaubey approximations to selected

approximate probability points of F', ^ -x N • (n,a,A;

(1) Upper entry for each (A,n) combination is the

sum of eight absolute errors (one error for each d

as d ranges over 4, 6, 8, 10, 20, 30, 40, 60).

(2) Lower entry for each CA,n) combination is the

maximiom absolute error selected from eight absolute

errors (one error for each d).

(3) Absolute error = |approximate percentage point -

exact percentage point|.

27

EH

f <

-d

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fa fa o

POINTS

UPPER 10%

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OXIJ

« 04 04 <

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<

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2.03

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ON

CHAPTER V

FOUR MOMENT APPROXIMATIONS TO THE NONCENTRAL DISTRIBUTIONS

Introduction

It is known that good (in the sense of simplicity and

accuracy) approximations to the noncentral x and noncentral

F can be obtained by using a Pearson curve with the correct

first four moments (e.g., see Johnson and Pearson, 1969).

2 That IS, the noncentral x inay be approximated by using a

type I Pearson curve (Beta distribution) which has the same

2 3 2 2 3-j(= 1 3/ 2 ^^^ ^2^~ 4/^2^ ^^ ^^® noncentral x distribution

of interest. The noncentral F may be approximated by using

a type VI Pearson curve (essentially a central F distribu­

tion) having the same 3-, and ^^ as the noncentral F dis­

tribution.

Indices of Skewness and Kurtosis and Moments Needed in the

Four Moment Approach

2 (1) Suppose X~X/ \\t then:

vn, A;

y = (n+A) y^ = CJ = 2(n+2A) (4.1)

y^ = 8(n+3A) y^ = 48(n+4A) -f 12 (n+2A)

3^ = 8(n+3A)^/(n-«-2A)^ 33 = 3 + 12 (n-i-4A) / (n-f2A) (2) If F*~F' -, . X , then:

(n,d,A) I = A/n

y = d(l+Jl)/(d-2)

M^ = a^=2d^(n+d-2) [l+2Jl+nJi^/(n+d-2) ]/[n(d-2)^(d-4) ]

34

35

M^ = 8d (n+d-2) (2n-i-d-2) [l-H3Jl+6nil /(2n-f-d-2)+2n JlV(n+d-2) (2n-Hd-2)]

/[n^(d-2)^(d-4)(d-6)]

^4 = 12d^(n+d-2) { [2(3n+d-2) (2n-Hd-2)-t-(n+d-2) (d-2) (n-H2)] x

(l+4il)+2n(3n+2d-4) (d-flO) il +4n (d+10) il -Hn (d-HlO) il /(n-fd-2) }

/[n^(d-2)'^(d-4) (d-6) (d-8)] (4.2)

2 3 2

^l - Vi3/y2 / 2 = y4/y2; using y^^, ^^ ^^^ ^4 above.

(3) If B~3/ V (B~Beta R.V. with parameters p and q) ,

then: y = p/(p-l-q) y2 = a^ = pq/[ (p+q) (p-<-q+l) ] (4.3)

(4) If F~F^^ ^ j , then:

y = d/(d-2) y2 = a^ = 2d^(n+d-2)/[n(d-2)^(d-4)]

(4.4)

General Approach to the Four Moment Method

In using the four moment approach one first calculates

3.. and $2 ^^r the distribution of interest. Next, tables

provided by Pearson and Hartley (1972), Johnson et al.

(1963), or Bouver et al. (1974) are entered with the calcu­

lated values of /3T and 32* The tables give standardized

percentage points of Pearson curves with the specified

/37 and 32' Thus, the corresponding approximate percentage

points for the distribution of interest are calculated by

using the correct mean and standard deviation of the dis­

tribution of interest. 2

For example, percentage points of a noncentral x with

particular n and A may be desired. The mean, variance, 3-j_,

36

2 and 3o of X/ -v N can be obtained by using relations C4.1).

Z vn , A; The Pearson curve tables are entered; and a value, say y, is obtained. The corresponding desired approximate per-

2 centage point for the noncentral x is given by

X = /yj y + y ,

where y2 and y are obtained from (4.1).

At first glance, the above procedure appears to in­

volve very little effort. One difficulty encountered is

that bilinear interpolation in the Pearson curve tables is

often necessary to obtain an approximate percentage point.

If many percentage points are desired, this procedure can

become rather time consuming and quite laborious. Another

problem in this procedure is that (3^, 32) points often

fall outside the range of the tables. A computer program

provided by Bouver and Bargman (1974) might be used to

defeat the above problems, but this program involves itera­

tive techniques making it somewhat unattractive.

To try to overcome the interpolation drawback and at

the same time obtain percentage points of the noncentral

distributions for a wide range of parameters, the efficient

and expedient subroutine CORFIS (Herring, 1974) is utilized

Recall that CORFIS will aid in generating percentage points

at the upper and lower 10%, 5%, and 2.5% levels. Also,

recall that the subroutine's CPU time per call is approxi­

mately 11 milliseconds (using double precision on an IBM

37

370/14 5). Thus, it is felt that CORFIS used in conjunction

with the four moment method provides an expedient and simple

method to obtain a wide variety of approximate percentage

points. The CORFIS-four moment method is also quite accur­

ate as will be shown subsequently.

Approach to the Four Moment Method Using CORFIS

The following steps are necessary to obtain percentage 2

points for \\ ^. i 2

(a) Compute y, y2f 3-,, 32 for x\^ ^\ according to relations (4.1).

2 (b) Since the (3 / 33) points for x[^ ^>^ fall in the type I region of the Pearson system, equate the cal-

2 culated 3-. and 32 of the noncentral x to 3-, and 32 of

a Beta distribution with parameters p and q. Solving

for p and q as in Elderton and Johnson (1969) one

obtains:

p = |{(r-2)-r(r-H2) [33_/(3i(r+2)^-H6(r-H))]^/^} + 1

q = |{(r-2)-Hr(r+2) [3-L/(6;L^''"^^^^"^^^^'''^^^^^^'^^^ ""

r = 6(32-3^-l)/(6-<-33^-232)

(c) Recall that if F~F,2p 2q) (central F R.V. and 2p

and 2q d.f.), then 3 = (2p)F/[2q+(2p)F] is a Beta R.V.

with parameters p and q. Thus, one should call the

subroutine CORFIS using 2p and 2q as n.d.f. and d.d.f.,

the major input parameters for CORFIS (see the intro-

38

duction. Chapter I, for an explanation of CORFIS input

parameters).

(d) CORFIS will return a percentage point, say f ,

for a central F distribution with 2p and 2q d.f. To

obtain the corresponding Beta (p,q) percentage point

use

^a = (2p)f^/[2q + (2p)f^] .

(e) Standardize 3^ with the mean and standard deviation a

of Beta(p,q) obtained from relations (4.3).

i.e., 3* = (3 -M)/O

2 (f) Finally, to obtain the desired x] -\\ percentage (n;A

point, X , use the following:

X = a3* + y , a a

where y and a are the mean and standard deviation of 2

X/ AX obtained from relations (4.1). ^(n,A)

The procedure to obtain percentage points of F', ^ »

is very similar to that outlined above for the noncentral

X . The n.d.f. and d.d.f. used to call CORFIS are:

n.d.f. = 2(r) (r- 2){33_/[3; (r-i-2) +16(r+l)]}- / + 2

d.d.f. = (r-2)+r(r+2){3 /[3;L '*" ^ " ^ ^ " ^ ^ " ' ^ "

r = 6(32-33^-l)/[6+33^-232] (4.5)

3-, and 3^ are obtained from relations (4.2).

39

After CORFIS returns the central F percentage point,

standardize the point using the mean and standard deviation

of a central F with parameters n.d.f. and d.d.f. given in

(4.5) (mean and standard deviation formulas are given in

relations 4.4). Finally, the percentage point of FJ , ,x (n , Cl, A;

is obtained by multiplying the standardized central F point

by the standard deviation of F' ^ ., and adding to this (n,d,A) ^

product the mean of F', -, , x (see relations (4.2) for mean (n,a,A)

and standard deviation formulas).

Summary of Results Using the CORFIS Four Moment Approach

For the four moment results, no general accuracy

analyses tables are presented as was done in the case of the

two and three moment fits. However, it is felt that by

using the previous accuracy tables (see tables 1-10); the

tables of selected approximate percentage points for all

methods; and the narrative presented in this section, one

can obtain an adequate knowledge of the general accuracy 2

of the four moment methods for both xL ^^ and FJ^ d,A)'

Noncentral F Results

Approximate percentage points (upper and lower 10%, 5%,

and 2.5%) of F' . ,x were generated by the CORFIS- four (n,a,A;

moment method for:

n = 1, 2, 3, 5, 10, 15, 30, 60

d = 10, 20, 30, 40, 60

A = 2, 6, 10, 14, 20 .

40

The approximate points were compared to percentage points

obtained from Lachenbruch (1967). The Lachenbruch points

were regarded as exact points (see Chapter III for Lachen­

bruch accuracy considerations).

For the upper level points, a comparison of the four

moment method with Tiku's (1965) three moment method was

carried out. These points might also have been compared

to those generated by Chaubey's (1976) method, but little

more insight would be gained since the Tiku and Chaubey

methods are very comparable in the upper tail. For all

cases considered in the upper level the four moment method

and Tiku's approximation are very comparable. At n = 1,

the four moment approach appears to be better. However,

in terms of percentage error, the difference between the

Tiku and four moment approach is not very significant.

In considering the lower percentage points, comparison

of the four moment method was made to the Tiku approximation

for:

(a) 10% level.

(b) 5% level for A = 14 with d = 10 and A = 20 with

d = 10, 20, 30.

(c) 2.5% level for A = 20 with d = 10.

In all other comparisons, the four moment method was com­

pared to the Chaubey approximation, since the Chaubey fit

is more accurate than Tiku's method for other than the

41

above parameter considerations. In general, it was found

that the four moment approximation is more accurate at

most all points of comparison. (There are quite a few

ties in accuracy, especially for large n of F', ^ -v N •) ^n,a, A;

The improvement in accuracy over the other two approxima­

tions is noted particularly for smaller n, say n = 1 or 2.

Parameter combinations at which the four moment method is

not the most accurate are noted as follows:

(1) lower 10%; A >_ 10 with n = 1, 2; Tiku is better.

(2) lower 2.5%; A = 6 with d = 6; Chaubey is better.

(3) lower 2.5%; A = 14 with d = 10, 20; Chaubey is

better.

In summary, it seems as if the four moment approach

is the best overall approximation of the approximations

considered. The major improvement is in the lower tail of F', J .X. However, it must be noted as n of FJ -, ,, in-(n,d,A) ^n,a,A;

creases, all of the approximations yield more and more

analogous results. A shortcoming of the four moment approxi­

mation is that it will not give percentage points if d of

P' , , is less than or equal to 10. For d £ 10, the (n,d,A)

Chaubey method may be used to generate lower points; and

the Tiku or Chaubey approximation may be used to generate

upper points. Tables 31-34 of the appendix show selected

approximate percentage points generated by the four moment

method.

42

2 Noncentral x Results

Approximate percentage points (upper and lower 10%, 2

5%, 2.5%) of Xjj ; j were obtained using the CORFIS - four

moment method for

n = 1(1)12, 15, 20

/A = .2(.2)6 .

The accuracy of the method was checked by comparing the

approximate point to exact points provided by Johnson

(1968). Furthermore, the four moment fit was compared to

the three moment Pearson (1959) approximation. Tables

35-40 of the appendix offer examples of selected approxi-

2 mate percentage points of x/' -v \ obtained by the four

n , A; moment method.

10% Level - x)^ ss ^(n,A)

With respect to 10% points, comparisons between Pear­

son 's method and the four moment approximation revealed

that the four moment fit is the better overall approxima­

tion.

In the upper tail, the four moment method is either

better or provides the same accuracy as the three moment

approximation. The differences in accuracy become less as

2 n of X/' NN increases. In terms of a comparison between

^(n. A)

the two approximations based on percentage error, the fol­

lowing graphic should give one as idea of the difference

in accuracy of the approximations in the upper tail.

Table 11

FOUR MOMENT .VS. THREE MOMENT FOR UPPER 10% - Y' (n. A)

*0% .Ox%

THREE MOMENT METHOD

.Ox%

.x%

OCCURS

OCCURS

DID NOT OCCUR FREQUENTLY

OCCURS

43

*percentage error = [|approx. pt. -exact pt.I/exact pt.] X 100

For example, the table indicates that if the three moment

percentage error is .Ox%, then one can expect the four

moment percentage error to be 0% ("x" is used to give one

a general idea of the magnitude of the error).

In the lower tail, the four moment approach was found

to be superior to the three moment method. The most

noticeable improvement in accuracy over the three moment

approximation given by the four moment fit is offered at

n = 1 or 2. Table 12 below offers an illustration of this

point.

For the upper 5% points, the four moment approximation

was determined to be the better approximation. Table 13

should provide insight into the major findings.

44

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45

Table 13

FOUR MOMENT .VS. THREE MOMENT FOR UPPER 5%

*0% .Ox%

,2 ^(n,A)

THREE .Ox%

MOMENT

METHOD .x%

OCCURS

DOES NOT OCCUR

dominant for A > 14

OCCURS

*percentage error = [| approx. pt. - exact pt. I/exact pt.J X 100 '

The same type interpretation for table 13 applies as was

given in the example of interpretation to table 11.

In the lower tail, the Pearson three moment approxi­

mation was found to be as good if not better than the four

moment fit for A 21.16. On the other hand, the four

moment approximation is generally better for A < 21.16,

particularly for n = 1 or 2.

2.5% Level - x', ,,)

The following table 14 should summarize the general

accuracy considerations for the upper 2.5% level.

TABLE 14

COMPARISONS OF THE FOUR AND THREE MOMENT FITS TO X\^ IN

THE UPPER TAIL

A RANGE BETTER APPROX. PERCENTAGE ERROR (4 MOMENT.VS. 3

MOMENT)

.04£A<_1.96

2.56£A£29.16

31.36<A<36

FOUR THREE

Approximately Equal; Three is better in some cases.

0% -VS- .Ox%

.Ox% -VS- 0%

46

In the lower 2,5% tail, the four moment approxima­

tion is superior to Pearson's fit for all parameter

combinations examined. The accuracy improvement is most

noticeable for n = 1 or 2. The accuracy considerations

for n = 1 or 2 are similar to those illustrated in table

12 for the lower 10% points.

In summary, the four moment method was found to be

the overall superior approximation for obtaining approxi-

2

mate percentage points to x / ^\• As shown in the narra­

tive above the method offers the greatest improvement in

accuracy over the three moment fit at the lower levels.

However, the same caution applies as in the case of the

three moment Pearson fit; the four moment method will

often yield negative points for small n and small A for

the lower probability levels.

General Summary of the Four Moment Approach

On the whole, the four moment method is a more accur-2

ate method to obtain percentage points of both xjj x) ^^^

F\ . ,x than the two and three moment methods presented (n,d,A)

earlier. The superiority of the method is noted mainly

for the lower level percentage points. Furthermore, the

four moment method is just as simple to use as the two and

three moment approximations. That is, the simplicity as­

pect of all the approximations is derived through the use

of CORFIS. Without the help of CORFIS, all of the approxi-

47

mations would be much harder to use because of interpola­

tion problems and finite range table limitations.

A Final Comment on the Four Moment Method

The CORFIS four moment method outlined in this chapter

can be used to obtain percentage points of distributions

for which the sample moments only are readily available.

All that is required is that the sample (3-, / 35) point

fall either into the type I region or the type VI region

of the Pearson system of curves.

For example, suppose the sample (3-,/ 32) point for

the distribution of the R.V. X falls in the type I region.

The procedure used to obtain percentage points using CORFIS

is outlined as follows:

** (See the procedure in this Chapter outlined under

Approach to the Four Moment Method Using CORFIS for the

analogous steps.)

(a)' Compute the appropriate sample moments and sample

3 and 3^. (Assume a sample of size n from X) 1 ^

2_ ? .. _-)Vn • = X = I x./n m = S = I (X.-1 i=l ^ i=l

n ^ ? - 4 m. = I (x.-x)Vn m, = I (x^-x) /n

^ i=l ^ i=l

3* = m3/m^ ^2 " ''4/"'2

48

(b)' Solve for p and q as outlined in (b)** with

3^ and ^^ substituted for 3, and 32*

(c)' - ( e ) ' : same steps as (c)-(e)**

(f) ' To obtain the desired percentage point of X, say

X , use the following: Uw

* _

X = S3 + X ,

a a

where x and S are given in step (a)'.

The procedure to obtain percentage points of a sampled

unknown distribution whose sample (3w 32) point falls into

the type VI region is similar to the procedure used for

F'. J -v N with appropriate modifications such as those of in , Q , A;

(a) ' - (f) '.

CHAPTER VI

AN EXAMPLE OF THE USE OF CORFIS OBTAINED NONCENTRAL

POINTS IN A MODIFIED SATTERTHWAITE PROCEDURE

Introduction

In this chapter it is desired to apply some of the

explored higher order moment fitting techniques (specific­

ally a three moment fit) in a novel way and at the same

time to present a situation where the CORFIS methods of

generating non-central F percentage points can be used.

Specifically, a modified Satterthwaite (1946) procedure is

derived which is used to help formulate a test for signifi­

cance of the fixed effect in the analysis of variance of

the three-factor mixed model (A-fixed, B and C random)

(Ostle, 1963). The derivation of the test is presented

in the sequel with appropriate emphasis given to the moment

fitting technique and the use of the CORFIS methods to

obtain a critical point for the test. No attempt is made

to determine the true size or power associated with the

derived test. Research on the usefulness and power of

the test will be conducted at a later time.

The Usual Satterthwaite Technique

In the usual Satterthwaite procedure one deals with

a linear combination

1=1

49

50

, where it is assumed the V. Ci=l,...,k) are independent

mean squares that are estimates of corresponding variance 2

components 9.. The V. are such that n. V . / S . - X A N for 1 1 1 1 1 '^(n.)

i=l, ..., K. Also one assumes the following:

^2^2 . „ . „ 2 E [ 1 7 1 = ^ ' where Y^Xf,

(i.e., f2L2/E[L2] is approximately distributed as a central 2

X with f2 d.f.)

The next step in the procedure is to equate the first two

moments of f2L2/E[L2] and Y and solve for f2. Upon solv­

ing the equations the following is obtained:

K r, K (a.e.)^ f = ( a e.)V[ I — V ^ ^ ^ -2 ^ i=l 1 ^ i=l ^i

Finally, the "best" estimates v. for 6. are substituted into 5.2 and one obtains the estimator

K r, K (a.v ) ^ f = ( I a.v )V[ I \^ ] . (5.3)

^ i=l ^ ^ i=l i The Modified Satterthwaite Technique

For the linear combination

K LT = y a.V. 1 i=l ^ ^

Assume the following:

(1) V., are independent mean squares.

(2) V is a noncentral mean square component. (Other

V. . could be assumed noncentral, but V., alone being 1' s ^

noncentral is adequate in the context of the mixed

51

model analysis of variance application to be presented).

This assumption implies Z. -^Xi -, s • n.V^ 2 1 (n^,A^)

^ ^ " ^ ^^(n^)' ^ = 2 K.

(4) f^L^/E(L^) = pX, where X-xJf ^s-

In this technique the first three moments of f,L^/ECL,)

and pX are equated. From these equations one obtains the

following:

p = {-4(y^ - yy3) + [8y^(2y^ - My^^)\^^'^}/2MM,^ 2

f^ = 2p(2-p)y /y2

A = (l-p)f^/p

, where y = E[L-j ] = a^0^(n^+A^)/n^ + \ a^e^ (5.4)

^ "^ M

2A2, . , X / 2 . r ^ 2.2 y = Var(L«,) = 2a^ ^ (n.,+2A,)/n, ^ \ 2a^eVn. 2 i. 1 1 1 1 1 . ^ ^ 1 1 1 1=2

y3 = E{[L -y]^} = 8a^e^ (n -H3A )/n + \ 8a^0^/n^ i=2

Estimates for p, f,, and A can be obtained by substituting

the appropriate estimates for 0^ into y, y2/ and y . (Note:

A., is specified according to hypothesis for the mixed model

example presented and 0-| = n^ ^i/f^i •*• i^) • These esti­

mates for y, y2/ and y are given by:

K y = y a.V. ^ . ^ T 1 1

1=1

^2 = 2a^v^(n^+2A^)/(n^-HA^) + I 2a^v^/n^ j^ ^ £*

^3 " 8a^v^(n^-<-3A^)/(n^+A^) + I 8a^v^/n^

52

Specific Application of the Modified Satterthwaite Approach

A specific application of the modified Satterthwaite

procedure can be found in the analysis of variance of the

three-factor mixed model (A-fixed, B and C random). It

is well known that an exact test for the significance of

factor A does not exist. There are several possibilities

for test statistics in this mixed model situation. Two

such statistics are presented, with the understanding that

similar procedures can be followed for the development of

other test statistics.

In general, in developing a test statistic, two linear

combinations of mean squares are considered (with no mean

square appearing in both combinations) whose expectations

are the same under the null hypothesis. The mean square

whose expectation is larger under the alternative hypothesis

is placed in the numerator. The resulting statistic is then

treated as a noncentral F statistic. If the usual Satter­

thwaite procedure is applied to both numerator and demon-

inator, the statistic would be treated as a central F

since the linear combinations are assumed to be composed

of central mean squares and the linear combinations are

assumed to be approximately distributed as central Chi-

Squares.

For the mixed model analysis of variance problem one

wishes to test:

53

^o * - ^ ^

vs.

H. : a > £

Two likely modified Satterthwaite test statistic

^1 = (V^/p^)/(V3+V2-V4)

F2 = [(V^-fV4)/p2]/(V2+V3)

s are

, where

V 1 ~ ^^A ("®^ square associated with factor A)

^2 = "^AB

^3 = "^AC

V. = MS^^^ 4 ABC

For illustration purposes, suppose F2 is selected as

the test statistic. In the notation of the modified Satter­

thwaite procedure L, = V,+V.. One would obtain estimates

for p2, f-., and A as outlined in the previous section on

the modification. Next, let the denominator of F^ be de­

noted by Lp = Vp+V-. The d.f., say f2, associated with

L2 are obtained by the usual Satterthwaite technique (see

relation 5.3). Thus, F2 would be treated as a noncentral

F statistic. The appropriate decision rule for this test

is:

54

^1 Re3ect H at a chosen a level, if - ^ > x , where

o PL2 o'

^o ^^ ^^^ appropriate percentage point of a noncentral

F with parameters f^, f2 and A.

The appropriate percentage point can be obtained by

using one of the previously presented approximations.

The CORFIS moment approximations offer quite accurate

approximate percentage points as shown previously. Also,

since f.., f2f and A are often non-integer, the CORFIS

methods are frequently preferred over limited parameter

range tables and other methods of generating points.

(CORFIS methods are valid for non-integer valued para­

meters of F' , , as well as integer valued.) (n,d,A)

Summary

In this chapter one example of the possible future

applications of higher order moment fitting techniques

is presented. The usefulness of the modified Satterth­

waite procedure has not been verified, but the technique

is worthy of future research. The Satterthwaite moment

fitting example also illustrates the main application of

higher order moment fitting techniques explored in this

thesis; namely, the generation of percentage points of

noncentral distributions.

APPENDIX

SELECTED APPROXIMATE PERCENTAGE POINTS

OF x;^,,) AHD F;^^^ ,)

USING CORFIS METHODS

55

TABLE 15

SELECTED APPROXIMATE UPPER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE THREE MOMENT PEARSON APPROXIMATION

56

DF^

1

5

8

10

12

15

2U

^IDA

5 . 1 5 0

- . 0 6 9 *

1 1 . 0 1

- . 0 2

1 4 . 9 9

- . 0 1

1 7 . 5 6

. 0 0

2 0 . 0 7

- . 0 1

2 3 . 7 8

. 0 0

2 9 . 8 2

. 0 0

4

1 0 . 6 8

- . 0 9

1 5 . 8 4

- . 0 3

1 9 . 5 7

- . 0 3

2 2 . 0 2

- . - 2

2 4 . 4 4

- . 0 2

2 8 . 0 4

- . 0 1

3 3 . 9 5

. 0 0

9

1 8 . 2 6

- . 0 7

2 3 . 0 7

- . 0 4

2 6 . 6 2

- . 0 4

2 8 . 9 8

- . 0 2

3 1 . 3 2

- . 0 2

3 4 . 8 0

- . 0 2

4 0 . 5 7

- . 0 1

16

2 7 . 8 4

- . 0 5

3 2 . 4 5

- . 0 5

• 3 5 . 8 9

- . 0 4

3 8 . 1 8

- . 0 3

4 0 . 4 5

- . 0 3

4 3 . 8 6

- . 0 2

4 9 . 5 0

- . 0 2

25

3 9 . 4 1

- . 0 5

4 3 . 9 1

- . 0 4

4 7 . 2 7

- . 0 3

4 9 . 5 0

- . 0 4

5 1 . 7 4

- . 0 2

5 5 . 0 8

- . 0 2

6 0 . 6 3

- . 0 2

36

5 2 . 9 8

- . 0 4

5 7 . 4 0

- . 0 3

6 0 . 7 0

- . 0 3

6 2 . 9 0

- . 0 3

6 5 . 1 0

- . 0 3

6 8 . 3 9

- . 0 4

7 3 . 8 7

- . 0 2

*Lower Entry is (Approximate Point - Exact Point).

57

TABLE 16

SELECTED APPROXIMATE LOWER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE THREE MOMENT PEARSON APPROXIMATION

N jAMDA ^

DF 1

5

8

10

12

15

20

- . 0 8 1 8 3

- . 1 2 4 5 *

1 . 9 5 3

- . 0 0 4

3 . 9 4 2

- . 0 0 1

5 . 3 6 6

. 0 0 0

6 . 8 4 1

. 0 0 0

9 . 1 2 6

. 0 0 0

1 3 . 0 7

. 0 0

4

. 5 4 4 4

. 0 0 2 7

3 . 2 7 6

. 0 0 5

5 . 4 9 3

. 0 0 5

7 . 0 2 2

. 0 0 5

8 . 5 8 0

. 0 0 3

1 0 . 9 6

. 0 0

1 5 . 0 3

. 0 0

9

2 . 9 9 5

. 0 4 2

6 . 1 1 9

. 0 2 3

8 . 5 2 5

. 0 1 7

1 0 . 1 5

. 0 1

1 1 . 7 9

. 0 1

1 4 . 2 8

. 0 0

1 8 . 4 9

. 0 1

16

7 . 4 2 9

. 0 3 9

1 0 . 7 7

. 0 3

1 3 . 3 0

. 0 2

1 5 . 0 0

. 0 2

1 6 . 7 1

. 0 2

1 9 . 2 9

. 0 2

2 3 . 6 2

. 0 1

25

1 3 . 8 6

. 0 3

1 7 . 3 3

. 0 2

1 9 . 9 5

. 0 2

2 1 . 7 1

. 0 3

2 3 . 4 6

. 0 2

2 6 . 1 1

. 0 2

3 0 . 5 4

. 0 2

36

2 2 . 2 9

. 0 3

2 5 . 8 6

. 0 3

2 8 . 5 4

. 0 3

3 0 . 3 3

. 0 3

3 2 . 1 2

. 0 2

3 4 . 8 2

. 0 2

7 3 . 8 7

.02

*Lower Entry is (Approximate Point - Exact Point)

TABLE 17

SELECTED APPROXIMATE UPPER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE TWO MOMENT PATNAIK APPROXIMATION

58

N DF

1

5

8

10

12

15

20

oAMDA,

5 . 0 7 8

- . 141*

1 1 . 0 1

- . 0 2

1 4 . 9 9

- ^ 0 1

1 7 . 5 6

. 00

2 0 . 0 7

- . 0 1

2 3 . 7 8

. 0 0

2 9 . 8 2

. 0 0

4

1 0 . 6 2

- . 1 5

1 5 . 8 4

- . 0 3

1 9 . 5 8

- . 0 2

2 2 . 0 3

- . 0 1

2 4 . 4 5

- . 0 1

2 8 . 0 4

- . 0 1

3 3 . 9 5

. 0 0

9

1 8 . 2 6

- . 0 7

2 3 . 0 9

- . 0 2

2 6 . 6 5

- . 0 1

2 9 . 0 0

. 0 0

3 1 . 3 4

. 00

3 4 . 8 2

. 0 0

4 0 . 5 8

. 0 0

16

2 7 . 8 8

- . 0 1

3 2 . 5 0

. 00

3 5 . 9 4

. 0 1

3 8 . 2 2

. 0 1

4 0 . 4 9

. 0 1

4 3 . 8 9

. 0 1

4 9 . 5 3

. 0 1

25

3 9 . 4 9

. 0 3

4 3 . 9 8

. 0 3

4 7 . 3 3

. 0 3

4 9 . 5 7

. 0 3

5 1 . 7 9

. 0 3

5 5 . 1 3

. 0 3

6 0 . 6 7

.02

36

5 3 . 0 8

. 0 6

5 7 . 4 9

.06

6 0 . 7 9

.06

6 2 . 9 8

. 0 5

6 5 . 1 8

. 0 5

6 8 . 4 7

. 04

7 3 . 9 3

.04

* Lower Entry is (Approximate Point - Exact Point)

59

TABLE 18

SELECTED APPROXIMATE LOWER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE TWO MOMENT PATNAIK APPROXIMATION

^ DF

1

5

8

10

12

lb

20

^MDA ^

.08266

.03996*

1.974

.017

3.951

.008

5.371

.005

6.845

.004

9.128

.002

13.07

.00

4

.8826

.3409

3.388

.1170

5.557

.069

7.068

.051

8.616

.039

10.99

.03

15.05

.02

9

3.347

.3940

6.310

.2140

8.658

.1500

10.26

.12

11.88

.10

14.35

.07

18.54 .06

16

7.768

.3780

11.00

.26

13.49

.21

15.16

.18

16.85

.16

19.40

.13

23.70

.09

25

14.18

.35

17.59

.28

20.17

.24

21.90

.22

23.64

.20

26.26

.17

30.66

.14

36

22.60

.34

26.12

.29

28.77

.26

30.54

.24

32.32

.22

34.99

.19

39.47

.17

* Lower Entry is (Approximate Point - Exact Point)

60

TABLE 19

SELECTED APPROXIMATE UPPER 2.5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE THREE MOMENT PEARSON APPROXIMATION

^

1

5

8

10

12

15

20

VMDA ,

8 . 6 7 3

- . 0 9 2 *

1 5 . 2 1

- . 0 1

1 9 . 6 3

- . 0 1

2 2 . 4 6

- . 0 1

2 5 . 2 3

. 0 0

2 9 . 2 9

. 0 0

3 5 . 8 6

. 0 0

4

1 5 . 6 4

- . 0 4

2 1 . 2 0

- . 0 1

2 5 . 2 4

. 0 0

2 7 . 8 8

. 0 0

3 0 . 4 9

. 0 0

3 4 . 3 6

. 0 0

4 0 . 7 0

. 0 0

9

2 4 . 5 9

- . 0 1

2 9 . 7 1

. 00

3 3 . 4 9

. 0 0

3 5 . 9 9

.00

3 8 . 4 8

. 0 0

4 2 . 1 8

. 00

4 8 . 3 0

. 0 0

16

3 5 . 5 3

. 0 1

4 0 . 3 9

.00

4 4 . 0 2

. 0 1

4 6 . 4 2

. 00

4 8 . 8 2

. 00

5 2 . 4 0

.00

5 8 . 3 4

.00

25

4 8 . 4 5

. 0 1

5 3 . 1 6

. 0 1

5 6 . 6 8

. 0 1

5 9 . 0 2

. 0 1

6 1 . 3 5

. 0 1

6 8 . 8 4

.00

7 0 . 6 5

. 0 1

36

6 3 . 3 7

. 0 1

6 7 . 9 7

. 0 1

7 1 . 4 1

.00

7 3 . 7 1

. 0 1

7 5 . 9 9

. 0 1

7 9 . 4 2

. 0 1

8 5 . 1 2

. 0 1

* Lower Entry is (Approximate Point - Exact Point).

61

TABLE 20

SELECTED APPROXIMATE LOWER 2.5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE THREE MOMENT PEARSON APPROXIMATION

DF| 1

5

8

10

12

15

20

LMDA ,

-.2184

-.2210*

.9827

-.0303

2.454

-.012

3.577

-.006

4.777

-.004

6.686

-.002

10.08

.00

4

-.3852

-.4362

1.648

-.1040

3.426

-.054

4.691

-.038

6.004

-.028

8.047

-.018

11.60

-.01

9

.8888

-.1942

3.497

-.1069

5.557

-.072

6.960

-.057

8.406

-.047

10.60

-.04

14.36

-.02

16

4.043

-.1190

6.991

-.085

9.250

-.067

10.78

-.05

12.31

-.06

14.65

-.04

18.59

-.03

25

9.158

-.084

12.32

-.07

14.72

-.05

16.32

-.06

17.94

-.05

20.38

-.04

24.48

-.03 1

36

16.26

-.06

19.56

-.06

22.06

-.05

23.73

-.04

25.40

0.04

27.92

-.04

32.14

-.04

* Lower Entry is (Approximate Point - Exact Point).

TABLE 21

SELECTED APPROXIMATE LOWER 10% POINTS OF F' ^ , , (n,d,A)

USING THE PATNAIK TWO MOMENT APPROXIMATION A = 6

62

\ d

n 1

I

i

i>

iO

ib

30

60

4

1.62

.31*

1.07

.107

.892

.054

.7484

.0186

.6402

.0033

.6024

.001

.5617

.0001

-5392

0

6

1.66

.34

1.11

.126

.9246

.0598

.7842

.0218

.6807

.0042

.6455

.0015

.6079

.0002

.5871

0

8

1.68

.35

1.124

.126

.9443

.0637

.8059

.0236

.7065

.0047

.6735

.0015

.6390

.0002

.6201

0

10

1.69

.35

1.137

.131

.9574

.0663

.8207

.0249

.7245

.005

.6935

.0017

.6619

.0003

.6448

.0001

20

1.72

.37

1.165

.140

.9873

.0723

.8553

.0284

.7687

.006

.7442

.0022

.7227

.0002

.7131

.0001

30

1.73

.37

1.18

.15

.999

.075

.869

.030

.787

.007

.766

.003

.750

0

.746

0

40

1.74

.38

1.18

.14

1.000

.072

.876

.031

.797

.007

.778

.003

.766

0

.766

0

60

1.75

.39

1.19

.15

1.01

.077

.833

.031

.807

.007

.791

.003

.784

0

.789

0

*Lower Entry is (Approximate Point - Exact Point).

TABLE 22

SELECTED APPROXIMATE LOWER 10% POINTS OF F' . ,. (n,d,A)

USING THE CHAUBEY TWO MOMENT APPROXIMATION A= 6

63

n 1

2

3

b

iU

ib

30

60

4

1.35

.04*

.973

.010

.842

.004

.7302

.0004

.6368

-.001

.6013

-.0001

.5616

0

.5392

0

6

1.37

.05

.995

.011

.8689

.0041

.7633

.0009

.6766

.0001

.6441

.0001

.6077

0

.5871

0

8

1.37

.04

1.008

.0106

.8849

.0043

.7834

.0011

.7019

.0001

.6720

0

.6389

.0001

.6201

0

10

1.38

.04

1.017

.011

.8956

.0045

.7969

.0011

.7196

.0001

.6919

.0001

.6617

.0001

.6447

0

20

1.39

.04

1.036

.011

.9199

.0049

.8286

.0017

.7628

.0001

.7421

.0001

.7225

0

.7131

.0001

30

1.39

.03

1.04

.01

.929

.005

.841

.002

.780

0

.763

0

.750

0

.746

0

40

1.40

.04

1.05

.01

.934

.006

.847

.002

.790

0

.775

0

.766

0

.766

0

60

1.40

.04

1.05

.01

.939

.006

.854

.002

.800

0

.788

0

.784

0

.789

0

*Lower Entry is (Approximate Point - Exact Point).

64

TABLE 23

SELECTED APPROXIMATE LOWER 10% POINTS OF F' ^ ^ j

USING THE TIKU THREE MOMENT APPROXIMATION A = 6

^ n 1

2

i

b

10

15

30

60

4

1.28

-.03*

.950

-.013

.8310

-.007

.7272

-.0026

.6364

-.0005

.6012

-.0002

.5616

0

.5392

°

6

1.31

-.01

.976

-.008

.8600

-.0048

.7605

-.0019

.6763

-.0002

.6440

0

.6077

0

.5871

0

8

1.33

0

.992

-.005

.8771

-.0035

.7808

-.0015

.7016

-.0002

.6719

-.0001

.6389

.0001

.6201

0

10

1.34

0

1.003

-.003

.8887

-.0024

.7946

-.0012

.7192

-.0003

.6918

0

.6617

.0001

.6447

0

20

1.37

.02

1.028

.003

.9156

.0006

.8270

.0001

.7626

-.0001

.742

0

.7225

0

.7131

.0001

30

1.38

.02

1.04

.01

.926

.002

.840

.001

.780

0

.763

0

.750

0

.746

0

40

1.38

.02

1.04

0

.931

.003

.846

.001

.790

0

.775

0

.766

0

.766

0

60

1.39

.03

1.05

.01

.937

.004

.853

.001

.800

0

.788

0

.784

0

.789

0

*Lower Entry is (Approximate Point - Exact Point)

65

TABLE 24 SELECTED APPROXIMATE UPPER 10% POINTS OF F',

(n ,d ,A) USING THE PATNAIK TWO MOMENT APPROXIMATION A = 2

^ n 1

2

3

5

10

15

30

60

4

1 3 . 1

- . 2 *

8 . 4 6

- . 0 5

6 . 9 0

- . 0 2

5 . 6 4 3

- . 0 0 6

4 . 6 9 9

0

4 . 3 8 5

. 0 0 1

4 . 0 7 2

. 0 0 1

3 . 9 1 6

0

6

1 0 . 5

- . 3

6 . 6 7 0

- . 0 6

5 . 3 6 9

- . 0 2 3

4 . 3 1 5

- . 0 0 5

3 . 5 1 9

0

3 . 2 5 3

. 0 0 1

2 . 9 8 7

. 0 0 1

2 . 8 5 5

. 0 0 1

3

9 . 5 0

- . 2 2

5 . 9 5 2

- . 0 5 6

4 . 7 5 1

- . 0 2 4

3 . 7 7 7

- . 0 0 7

3 . 0 3 9

0

2 . 7 9 1

. 0 0 1

2 . 5 4 2

. 0 0 1

2 . 4 1 7

0

10

8 . 9 4

- . 2 1

5 . 5 6 4

- . 0 5 7

4 . 4 1 9

- . 0 2 2

3 . 4 8 8

- . 0 0 6

2 . 7 8 0

0

2 . 5 4 0

0

2 . 2 9 9

0

2 . 1 7 7

0

20

7 . 9 4

- . 2 0

4 . 8 7 6

- . 0 5 4

3 . 8 3 1

- . 0 2 2

2 . 9 7 6

- . 0 0 6

2 . 3 1 5

- . 0 0 1

2 . 0 8 8

0

1 . 8 5 4

. 0 0 1

1 . 7 3 3

. 0 0 1

30

7 . 6 4

- . 1 9

4 . 6 7

- . 0 5

3 . 6 6

- . 0 2

2 . 8 2

- . 0 1

2 . 1 7

- . 0 1

1 . 9 5

0

1 . 7 1

0

1 . 5 9

0

40

7 . 5 0

- . 1 9

4 . 5 7

- . 0 5

3 . 5 7

- . 0 2

2 . 7 5

0

2 . 1 1

0

1 . 8 8

0

1 . 6 4

0

1 . 5 2

0

60

7 . 3 6

- . 1 8

4 . 4 8

- . 0 4

3 . 4 9

- . 0 2

2 . 6 8

0

2 . 0 4

0

1 . 8 1

0

1 . 5 7

0

1 . 4 4

0

•Lower Entry is (Approximate Point - Exact Point)

66

TABLE 25

SELECTED APPROXIMATE UPPER 10% POINTS OF F', ^ , , (n,d,A)

USING THE CHAUBEY TWO MOMENT APPROXIMATION A = 2 ^

1

2

3

5

10

15

30

60

4

1 3 . 3

0*

8 . 5 2

. 0 1

6 . 9 2

0

5 . 6 5 0

- 0 0 1

4 . 7 0 0

. 0 0 1

4 . 3 8 5

. 0 0 1

4 . 0 7 2

. 0 0 1

3 . 9 1 6

0

6

1 0 . 7

- . 1

6 . 7 3

0

5 . 3 9 1

- . 0 0 1

4 . 3 2 1

. 0 0 1

3 . 5 2 0

. 0 0 1

3 . 2 5 3

. 0 0 1

2 . 9 8 7

. 0 0 1

2 . 8 5 5

. 0 0 1 '

8

9 . 6 7

- . 0 5

5 . 9 9 9

- . 0 0 9

4 . 7 7 0

- . 0 0 5

3 . 7 8 2

- . 0 0 2

3 . 0 3 9

0

2 . 7 9 1

. 0 0 1

2 . 5 4 2

. 0 0 1

2 . 4 1 7

0

10

9 . 1 0

- . 0 5

5 . 6 0 6

- . 0 1 5

4 . 4 3 6

- . 0 0 5

3 . 4 9 3

- . 0 0 1

2 . 7 8 0

0

2 . 5 4 0

0

2 . 2 9 9

0

2 . 1 7 7

0

20

8 . 0 6

- . 0 8

4 . 9 0 7

- . 0 2 3

3 . 8 4 3

- . 0 1

2 . 9 7 9

- . 0 0 3

2 . 3 1 6

0

2 . 0 8 8

0

1 . 8 5 4

. 0 0 1

1 . 7 3 3

. 0 0 1

30

7 . 7 5

- . 0 8

4 . 7 0

- . 0 2

3 . 6 6 0

- . 0 2

2 . 8 2

- . 0 1

2 . 1 7

- . 0 1

1 . 9 5

0

1 . 7 1

0

1 . 5 9

0

40

7 . 6 0

- . 0 9

4 . 6 0

- . 0 2

3 . 5 8

- . 0 1

2 . 7 5

0

2 . 1 1

0

1 . 8 8

0

1 . 6 4

0

1 . 5 2

0

60

7 . 4 6

- . 0 8

4 . 5 0

- . 0 2

3 . 5 0

- . 0 1

2 . 6 8

0

2 . 0 4

0

1 . 8 1

0

1 - 5 7

0

1 . 4 4

0

•Lower Entry is (Approximate Point - Exact Point)

TABLE 26

SELECTED APPROXIMATE UPPER 10% POINTS OF F', , , , (n,d,A)

USING THE TIKU THREE MOMENT APPROXIMATION A = 2

67

\ d n

1

2

3

5

10

15

30

60

4

1 3 . 3

0*

8 . 5 3

. 0 2

6 . 9 2

0

5 . 6 5

. 0 0 1

4 . 7 0

- 0 0 1

4 . 3 8 b

. 0 0 1

4 . 0 7 2

. 0 0 1

3 . 9 1 6

0

6

1 0 . 8

0

6 . 7 3

0

5 . 3 9 2

0

4 . 3 2 1

. 0 0 1

3 . 5 2 0

. 0 0 1

3 . 2 5 3

. 0 0 1

2 . 9 8 7

. 0 0 1

2 . 8 5 5

- 0 0 1

8

9 . 6 8

- . 0 4

6 . 0 0 2

- . 0 0 6

4 . 7 7 1

- , 0 0 4

3 . 7 8 3

- . 0 0 1

3 . 0 3 9

0

2 . 7 9 1

. 0 0 1

2 . 5 4 2

. 0 0 1

2 - 4 1 7

0

10

9 . 1 0

- . 0 5

5 . 6 0 8

- . 0 1 3

4 . 4 3 7

- . 0 0 4

3 . 4 9 3

- . 0 0 1

2 . 7 8 0

0

2 . 5 4 0

0

2 . 2 9 9

0

2 . 1 7 7

0

20

8 . 0 7

- . 0 7

4 . 9 0 7

- . 0 2 3

3 . 8 4 3

- . 0 1

2 . 9 7 9

- . 0 0 3

2 . 3 1 6

0

2 . 0 0 8

0

1 . 8 5 4

. 0 0 1

1 . 7 3 3

. 0 0 1

30

7 . 7 5

- . 0 8

4 . 7 0

- . 0 2

3 . 6 6

- . 0 2

2 . 8 2

- . 0 1

2 . 1 7

- . 0 1

1 . 9 5

0

1 . 7 1

0

1 . 5 9

40

7 . 6 0

- . 0 9

4 . 6 0

- . 0 2

3 . 5 8

- . 0 1

2 . 7 5

0

2 . 1 1

0

1 . 8 8

0

1 . 6 4

0

1 . 5 2

0

60

7 . 4 6

- . 0 8

4 - 5 0

- - 0 2

3 -50

- . 0 1

2 . 6 8

0

2 . 0 4

0

1 -81

0

1 .57

0

1 . 4 4

0

* Lower Entry is (Approximate Point - Exact Point)

TABLE 27

SELECTED APPROXIMATE LOWER 5% POINTS OF F' (n,d,A)

USING THE CHAUBEY TWO MOMENT APPROXIMATION A=14

68

n 1

.2

3

b

iU

ib

30

60

4

3.60

.07*

2.04

.02

1.52

.01

1.102

.005

.7800

.0008

.6686

.0003

.5519

.0004

.4894

-.0002

6

3.81

.06

2.18

.03

1.631

.014

1.190

.005

.8530

.0011

.7364

.0002

.6142

-.0001

.5486

.0001

9

3.94

.07

2.258

.025

1.697

.013

1.245

.005

.9001

.001

.7812

.0003

.6566

.0001

.5897

0

10

4.02

.06

2.311

.022

1.741

.012

1.283

.005

.9334

.0008

.8134

.0002

.6880

.0001

.6207

.0001

20

4.19

.03

2.430

.013

1.843

.007

1.372

.002

1.017

0

.8966

-.0001

.7730

-.0001

.7080

.0001

30

4.25

.02

2.47

0

1.88

0

1.41

0

1.05

0

.933

0

.812

0

.751

0

40

4.28

.01

2.50

.01

1.90

0

1.43

.01

1.07

"

.953

0

.835

0

.777

0

60

4.31

0

2.52

0

1.92

0

1.45

0

1.09

0

.975

0

.861

-.0001

.807

-.001

*Lower Entry is (Approximate Point - Exact Point)

TABLE 28

SELECTED APPROXIMATE LOWER 5% POINTS OF F' ^ ^ ^ j

USING THE TIKU THREE MOMENT APPROXIMATION A= 14

69

^ ' v ^

1

2

3

5

10

15

30

60

4

3.51

-.02*

2.01

-.01

1.50

-.01

1.095

--002

.7785

-.0007

.6681

-.0002

.5518

.0003

.4894

-.0002

6

3-72

--03

2-14

-.01

1.611

-.006

1.182

--003

.8512

--0007

-7359

-.0003

.6141

-.0002

.5486

.0001

8

3.85

-.02

2.222

-.011

1.677

-.007

1.237

-.003

,8981

-.001

.7805

-.0004

.6566

.0001

.5897

0

10

3.93

-.03

2.276

-.013

1.722

-.007

1.274

-.004

.9314

-.0012

.8127

-.0005

.6879

0

.6207

.0001

20

4.13

-.03

2.403

-.014

1.827

-.009

1.365

-.005

1.015

-.002

.8959

-.0008

.7729

-.0002

.7080

.0001

30

4.20

-.03

2.45

-.02

1.87

-.01

1.40

-.01

1.05

0

.932

-.001

.812

0

-751

0

40

4-24

-.03

2.48

-.01

1.89

-.01

1.42

0

1.07

0

.952

-.001

.835

0

.777

0

60

4.28

-.03

2.51

-.01

1.91

-.01

1.44

-.01

1.09

0

.974

-.001

.861

-.001

.807

-.001

*Lower Entry is (Approximate Point - Exact Point).

TABLE 29

70

SELECTED APPROXIMATE UPP?R 2.5% POINTS OF F' ^ ,, (n,d,A)

USING THE TIKU THREE MOMENT APPROXIMATION A = 20 n ^ ^

1

2

3

5

10

15

30

60

4

1 8 5

0*

9 6 . 6

. 2

6 7 . 1

. 2

4 3 - 4 7

. 0 7

2 5 - 7 9

. 0 2

1 9 - 9 1

. 0 3

1 4 . 0 5

- . 0 1

1 1 . 1 4

- . 0 1

6

114

0

5 9 . 3

. 1

4 1 . 1 1

. 0 6

2 6 . 5 4

. 0 6

1 5 . 6 3

. 0 3

1 2 . 0 0

. 0 2

8 . 3 9 7

0

6 - 6 1 1

-004

8

8 9 - 3

0

4 6 - 4

- 0 3

3 2 - 1 0

-02

2 0 . 6 6

. 0 2

1 2 . 1 0

0

9 . 2 6 0

. 0 0 6

6 . 4 3 5

- 0 0 1

5 . 0 3 9

- . 0 0 3

10

7 7 . 1

. 1

3 8 . 9 9

. 0 1

2 7 . 6 3

0

1 7 . 7 5

. 0 1

1 0 . 3 6

. 0 2

7 . 8 9

. 0 0 4

5 . 4 6 0

-004

4 . 2 5 6

. 0 0 6

20

5 7 . 0

0

2 9 . 4 9

0

2 0 . 3 2

0

1 2 . 9 8

0

7 . 4 8 7

- . 0 0 3

5 - 6 6 2

. 0 0 1

3 . 8 4 8

. 0 0 2

2 . 9 5 3

0

30

5 1 . 4

0

2 6 . 5

- . 1

1 8 . 3

0

1 1 . 6

0

6 . 6 8

0

5 . 0 3

0

3 . 3 9

. 0 1

2 . 5 7

0

40

4 8 - 8

0

2 5 - 2

0

1 7 . 3

0

1 1 . 0

0

6 . 2 9

0

4 . 7 3

0

3 . 1 6

0

2 . 3 9

0

60

4 6 . 2

0

2 3 . 8

0

1 6 . 4

0

1 0 . 4

0

5 . 9 2

0

4 . 4 3

0

2 . 9 5

0

2 . 2 1

0

* Lower Entry is (Approximate Point - Exact Point)

TABLE 30

SELECTED APPROXIMATE UPPER 2.5% POINTS OF F' ^ , , (n,d,A)

USING THE CHAUBEY TWO MOMENT APPROXIMATION A = 20

71

n ^

1

2

3

5

10

15

30

60

4

185

0*

9 6 . 6

. 2

6 7 . 1

. 2

4 3 . 4 6

. 0 6

2 5 . 7 9

. 0 2

1 9 . 9 1

. 0 3

1 4 . 0 5

- . 0 1

1 1 - 1 4

- . 0 1

6

114

0

5 9 . 3

. 1

4 1 . 1 1

. 0 6

2 6 . 5 4

. 0 6

1 5 . 6 3

. 0 3

1 2 . 0 0

- 0 2

8 . 3 9 7

0

6 - 6 1 1

. 0 0 4

8

8 9 . 3

0

4 6 . 4 1

- 0 4

3 2 - 1 0

. 0 2

2 0 . 6 7

. 0 3

1 2 . 1 0

0

9 . 2 6 1

. 0 0 7

6 . 4 3 5

. 0 0 1

5 . 0 3 9

- . 0 0 3

10

7 7 . 1

. 1

4 0 . 0 0

. 0 2

2 7 . 6 4

. 0 1

1 7 . 7 6

. 0 2

1 0 . 3 6

. 0 2

7 . 9 0 0

. 0 0 5

5 . 4 6 0

. 0 0 4

4 . 2 5 6

-006

20

5 7 . 1

. 1

2 9 . 5 2

. 0 3

2 0 . 3 3

. 0 1

1 2 . 9 9

. 0 1

7 . 4 8 9

- . 0 0 1

5 . 6 6 3

. 0 0 2

3 . 8 4 8

. 0 0 2

2 . 9 5 3

0

30

5 1 . 5

- 1

2 6 . 6

0

1 8 . 3

0

1 1 . 6

0

6 . 6 8

0

5 . 0 3

0

3 . 3 9

. 0 1

2 - 5 7

0

40

4 8 . 8

0

2 5 . 2

0

1 7 . 3

0

1 1 - 0

0

6 . 3 0

. 0 1

4 . 7 3

0

3 - 1 6

0

2 . 3 9

0

60

4 6 . 2

0

2 3 . 8

0

1 6 . 4

0

1 0 . 4

0

5 . 9 2

0

4 . 4 3

0

2 . 9 5

0

2 . 2 1

0

*Lower Entry is (Approximate Point - Exact Point)

72

TABLE 31

SELECTED APPROXIMATE LOWER 10% PTS OF F' • (n,d,A)

USING THE FOUR MOMENT APPROACH; A = 6.0

n

1

2

3

b

10

15

30

60

10

1.35

.01*

1.008

.002

.8912

.0001

.7956

-.0002

.7194

-.0001

.6918

0.0000

.6617

.0001

.6447

.0000

20

1.40

.05

1.039

.014

.9214

.0064

.8292

.0023

.7630

.0003

.7422

.0002

.7225

.0000

.7131

.0001

30

1.41

.05

1.050

.020

.932

.008

.842

.003

.781

.001

.763

.000

.750

.000

.746

.000

40

1.42

.06

1.05

.010

.937

.009

.848

.003

.790

.000

.775

.000

.766

.000

.766

.000

60

1.42

.06

1.06

.02

.943

.010

.855

.003

.800

.000

.788

.000

.784

.000

.789

.000

*Lower Entry is (Approximate Point - Exact Point)

73

TABLE 32

SELECTED APPROXIMATE UPPER 10% PTS. OF F' (n,d,A)

USING THE FOUR MOMENT APPROACH; A = 2.0

L ^ 1

2

3

5

10

ib

30

60

10

9.10

-.05

5.608

-.013

4.437

-.004

3.493

-.001

2.78

.00

2.54

.00

2.299

. 000

2.177

.000

20

8.09

-.05

4.915

-.015

3.847

-.006

2.981

-.001

2.316

.000

2.088

.000

1.854

.001

1.733

.001

30

7.79

-.04

4.71

-.01

3.67

-.01

2.83

.00

2.17

-.01

1.95

.00

1.71

.00

1.59

.00

40

7.64

-.05

4.61

-.01

3.59

.00

2.75

.00

2.11

.00

1.88

.00

1.64

.00

1.52

.00

60

7.50

-.04

4.52

.00

3.50

-.01

2.68

.00

2.04

.00

1.81

.00

1.57

.00

1.44

.00

*Lower Entry is (Approximate Point - Exact Point)

TABLE 33

SELECTED APPROXIMATE LOWER 5% PTS OF F' • (n,d,A)

USING THE FOUR MOMENT APPROACH; A = 14

74

n 1

2

3

5

10

15

30

60

' 1 0

3.94

-.02*

2.280

-.009

1.724

-.005

1.275

-.003

.9318

-.0008

.8129

-.0003

.6880

.0001

.6207

.0001

20

4.16

0.00

2.416

-.001

1.836

.000

1.369

-.001

1.016

-.001

.8964

-.0003

.7730

-.0001

.7080

.0001

30

4.24

.01

2.47

.00

1.88

.00

1.41

.00

1.05

.00

.933

.00

.812

.000

.751

.000

40

4.28

.01

2.50

.01

1.90

.00

1.43

.01

1.07

.00

.953

.000

.835

.000

.777

.000

60

4.33

.02

2.53

.01

1.93

.01

1.45

.00

1.09

.00

.975

.000

.861

-.001

.807

-.001

*Lower Entry is (Approximate Point - Exact Point)

75

TABLE 34

SELECTED APPROXIMATE UPPER 2.5% PTS. OF F '

USING THE FOUR MOMENT APPROACH; A = 20 (n,d,A)

^

n 1

2

3

5

iU

lb

30

60

10

77.1

.1*

39.99

.01

27.63

.00

17.75

.01

10.36

.02

7.900

.005

5.460

.004

4.256

.006

20

57.0

.0

29.50

.01

20.32

.00

12.98

.00

•7.488

-.002

5.662

.001

3.848

.002

2.953

.000

30

51.4

.0

26.6

.0

18.3

.0

11.6

.0

6.68

.00

5.03

.00

3.39

.01

2.57

.00

40

48.8

.0

25.2

.0

17.3

.0

11.0

.0

6.29

.00

4.73

.00

3.16

.00

2.39

.00

60

46.2

.0

23.8

.0

16.4

.0

10.4

.0

5.92

.00

4.43

.00

2.95

.00

2.21

.00

*Lower Entry is (Approximate Point - Exact Point)

76

TABLE 35

SELECTED APPROXIMATE LT>PER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE FOUR MOMENT METHOD

DF

1

5

8

10

12

15

20

\MDA ^

5 . 1 8 7

- . 0 3 2 *

1 1 . 0 2

- . 0 1

1 5 . 0 0

. 0 0

1 7 . 5 6

. 0 0

2 0 . 0 8

. 0 0

2 3 . 7 8

. 0 0

2 9 . 8 2

. 0 0

4

1 0 . 7 6

- . 0 1

1 5 . 8 7

. 0 0

1 9 . 5 9

- . 0 1

2 2 . 0 4

. 0 0

2 4 . 4 6

. 0 0

2 8 . 0 5

. 0 0

3 3 . 9 5

. 0 0

9

1 8 . 3 4

- . 0 1

2 3 . 1 2

. 0 1

2 6 . 6 6

. 00

2 9 . 0 1

. 0 1

3 1 . 3 4

. 0 0

3 4 . 8 2

. 0 0

4 0 . 5 8

. 0 0

16

2 7 . 9 0

. 0 1

3 2 . 5 0

. 0 0

3 5 . 9 3

.00

3 8 . 2 1

. 00

4 0 . 4 8

. 00

4 3 . 8 8

.00

2 3 . 6 1

.00

25

3 9 . 4 6

. 00

4 3 . 9 5

. 0 0

4 7 . 3 1

. 0 1

4 9 . 5 4

.00

5 1 . 7 7

. 0 1

5 5 . 1 0

.00

6 0 . 6 5

. 00

36

5 3 . 0 3

. 0 1

5 7 . 4 4

. 0 1

6 0 . 7 4

. 0 1

6 2 . 9 4

. 0 1

6 5 . 1 3

. 00

6 8 . 4 2

- . 0 1

7 3 . 8 9

. 00

* Lower Entry is (Approximate Point - Exact Point).

77

TABLE 36

SELECTED APPROXIMATE LOWER 10% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE POUR MOMENT METHOD

\LAMDA ^

4 > rH

5

8

10

12

15

20

. 0 0 9 9 4 7

- . 0 5 2 6 5

1 . 9 5 7

. 0 0 0

3 . 9 4 3

. 0 0 0

5 . 3 6 6

. 0 0 0

6 . 8 4 1

. 0 0 0

9 . 1 2 6

. 0 0 0

1 3 . 0 7

. 0 0

4

. 5 9 5 8

. 0 5 4 1

3 . 2 8 3

. 0 1 2

5 . 4 9 4

. 0 0 6

7 . 0 2 1

. 0 0 4

8 . 5 7 9

. 0 0 2

1 0 . 9 6

. 0 0

1 5 . 0 3

. 0 0

9

3 . 0 0 5

. 0 5 2

6 . 1 1 8

. 0 2 2

8 . 5 2 1

. 0 1 3

1 0 . 1 5

. 0 1

1 1 . 7 9

. 0 1

1 4 . 2 8

. 00

1 8 . 4 9

. 0 1

16

7 . 4 2 2

. 0 3 2

1 0 . 7 6

. 0 2

1 3 . 2 9

. 0 1

1 4 . 9 9

. 0 1

1 6 . 7 0

. 0 1

1 9 . 2 8

. 0 1

2 3 . 6 1

.00

25

1 3 . 8 5

. 0 2

1 7 . 3 2

. 0 1

1 9 . 9 4

. 0 1

2 1 . 7 0

. 0 2

2 3 . 4 5

. 0 1

2 6 . 1 0

. 0 1

6 0 . 6 5

. 00

36

2 2 . 2 8

. 0 2

2 5 . 8 4

. 0 1

2 8 . 5 2

. 0 1

3 0 . 3 1

. 0 1

3 2 . 1 1

. 0 1

3 4 . 8 0

. 0 0

3 9 . 3 1

. 0 1

*Lower Entry is (Approximate Point - Exact Point).

78

TABLE 37

SELECTED APPROXIMATE UPPER 5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE FOUR MOMENT METHOD

N^AMDA^

DF ' 1

5

8

10

12

15

20

6 . 9 9 5

- . 0 0 7

1 3 . 1 7

. 0 0

1 7 . 3 8

- . 0 1

2 0 . 0 9

. 0 0

2 2 . 7 4

. 0 0

2 6 . 6 4

. 0 0

3 2 . 9 6

- . 0 1

4

1 3 . 3 1

. 0 3

1 8 . 6 3

. 0 0

2 2 . 5 2

. 0 0

2 5 . 0 7

. 0 0

2 7 . 5 9

. 0 0

3 1 . 3 3

. 0 0

3 7 . 4 7

. 0 0

9

2 1 . 6 0

. 0 3

2 6 . 5 5

. 0 2

3 0 . 2 2

. 0 1

3 2 . 6 5

. 0 1

3 5 . 0 6

. 00

3 8 . 6 6

. 00

4 4 . 6 1

. 00

16

3 1 . 8 9

. 0 3

3 6 . 6 2

. 0 1

4 0 . 1 6

. 0 1

4 2 . 5 0

. 00

4 4 . 8 4

.00

4 8 . 3 4

.00

5 4 . 1 4

.00

25

4 4 . 1 7

. 0 2

4 8 . 7 8

. 0 2

5 2 . 2 2

. 0 1

5 4 . 5 1

. 0 1

5 6 . 7 9

. 0 1

6 0 . 2 1

.00

6 5 . 9 0

. 0 1

36

5 8 . 4 6

. 0 2

6 2 . 9 7

. 0 1

6 6 . 3 5

. 0 1

6 8 . 5 9

. 0 0

7 0 . 8 4

. 0 1

7 4 . 2 0

. 00

7 9 . 8 0

. 0 1

*Lower Entry is (Approximate Point-Exact Point).

79

TABLE 38

SELECTED APPROXIMATE LOWER 5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE FOUR MOMENT METHOD

V LAMDA ^

DF 1

5

1 8

10

1 ?

15

20

. 0 7 4 8 8

- . 0 8 5 5 5 *

1 . 3 9 1

- . 0 0 3

3 . 0 8 9

- . 0 0 1

4 . 3 4 7

. 0 0 0

5 . 6 7 2

- . 0 0 1

7 . 7 5 4

. 0 0 0

1 1 . 4 0

. 0 0

4

. 1 1 5 1

- . 0 6 5 7

2 . 3 7 5

- . 0 0 1

4 . 3 3 5

. 0 0 1

5 . 7 1 5

. 0 0 1

7 . 1 3 7

. 0 0 1

9 . 3 3 4

. 0 0 0

1 3 . 1 2

. 0 0

9

1 .866

. 0 2 9

4 . 6 7 4

. 0 1 2

6 . 8 8 1

. 0 0 8

8 . 3 8 6

. 0 0 6

9 . 9 1 5

. 0 0 5

1 2 . 2 4

. 0 0

1 6 . 2 1

. 0 1

16

5 . 5 7 3

. 026

8 . 6 8 7

. 0 1 5

1 1 . 0 7

. 0 1

1 2 . 6 7

. 0 1

1 4 . 2 9

. 0 1

1 6 . 7 3

. 0 1

2 0 . 8 5

. 0 1

25

1 1 . 2 8

. 0 2

1 4 . 5 8

. 0 2

1 7 . 0 7

. 0 1

1 8 . 7 5

. 0 1

5 6 . 7 9

. 0 1

2 2 . 9 6

. 0 1

2 7 . 2 1

. 0 1

36

1 8 . 9 8

. 0 1

2 2 . 4 0

. 0 1

2 4 . 9 8

. 0 1

2 6 . 7 1

. 0 1

2 8 . 4 4

. 0 1

3 1 . 0 4

. 0 1

3 5 . 3 9

. 0 1

*Lower Entry is (Approximate Point - Exact Point).

80

TABLE 39

SELECTED APPROXIMATE UPPER 2.5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE FOUR MOMENT METHOD

\LAMDA^

DF 1

5

8

10

12

15

20

8 . 7 9 4

. 0 2 9 *

1 5 . 2 2

. 0 0

1 9 . 6 4

. 0 0

2 2 . 4 7

. 0 0

2 5 . 2 3

. 0 0

2 9 . 2 9

. 0 0

3 5 . 8 6

. 0 0

4

1 5 . 7 4

. 0 6

2 1 . 2 3

. 0 2

2 5 . 2 5

. 0 1

2 7 . 8 9

. 0 1

3 0 . 5 0

. 0 1

3 4 . 3 6

. 0 0

4 0 . 7 0

. 0 0

9

2 4 . 6 4

.04

2 9 . 7 3

. 0 2

3 3 . 5 0

. 0 1

3 6 . 0 0

. 0 1

3 8 . 4 9

. 0 1

4 2 . 1 9

. 0 1

4 8 . 3 0

. 0 0

16

3 5 . 5 5

. 0 3

4 0 . 4 1

. 0 2

4 4 . 0 3

. 0 2

4 6 . 4 3

. 0 1

4 8 . 8 3

. 0 1

5 2 . 4 1

. 0 1

5 8 . 3 4

. 00

25

4 8 . 4 6

. 0 2

5 3 . 1 7

. 0 2

5 6 . 6 8

. 0 1

5 9 . 0 2

. 0 1

6 1 . 3 5

. 0 1

6 4 . 8 5

. 0 1

7 0 . 6 5

. 0 1

36

6 3 . 3 8

. 0 2

6 7 . 9 7

. 0 1

7 1 . 4 2

. 0 1

7 3 . 7 1

. 0 1

7 5 . 9 9

. 0 1

7 9 . 4 2

. 0 1

8 5 . 1 2

. 0 1

*Lower Entry is (Approximate Point - Exact Point).

81

TABLE 40

SELECTED APPROXIMATE LOWER 2.5% POINTS OF THE NONCENTRAL

CHI-SQUARE USING THE FOUR MOMENT METHOD

N;iAMDA ,

DF 1

5

8

10

12

15

20

- . 0 9 8 5 4 - . 1 0 1 2 *

1 . 0 0 6

- . 0 0 7

2 . 4 6 4

- . 0 0 2

3 . 5 8 2

- . 0 0 1

4 . 7 8 1

. 0 0 0

6 . 6 8 8

. 0 0 0

1 0 . 0 8

. 0 0

4 - . 1 7 6 7

- . 2 2 7 7

1 .729

- . 0 2 3

3 . 4 7 3

- . 0 0 7

4 , 7 2 5

- . 0 0 4

6 . 0 2 9

- . 0 0 3

8 . 0 6 4

- . 001

1 1 . 6 1

. 0 0

9 1 .050

- . 0 3 3

3 . 5 9 2

- . 0 1 1

5 . 6 2 4

- . 0 0 5

7 . 0 2 3

- . 0 0 3

8 . 4 5 1

- . 0 0 2

1 0 . 6 4

. 0 0

1 4 . 3 8

. 00

16 4 . 1 6 1

- . 0 0 1

7 . 0 7 6

. 0 0 0

9 . 3 1 8

. 0 0 1

1 0 . 8 3

.00

1 2 . 3 7

.00

1 4 . 6 9

.00

1 8 . 6 2

.00

25 9 . 2 4 7

. 0 0 5

1 2 . 3 9

. 00

1 4 . 7 8

. 0 1

1 6 . 3 8

. 00

1 7 . 9 9

.00

2 0 . 4 2

. 00

2 4 . 5 1

. 00

36 1 6 . 3 3

. 0 1

1 9 . 6 2

. 0 0

2 2 . 1 1

. 00

2 3 . 7 8

. 0 1

7 5 . 9 9

. 0 1

7 9 . 4 2

. 0 1

8 5 . 1 2

. 0 1

*Lower Entry is (Approximate Point - Exact Point).

REFERENCES

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Elderton, W. P. and Johnson, N. L. (1969) . Systems of Fre­quency Curves. Cambridge: Cambridge University Press.

Herring, T. A. (1974) . An Empirical Modification of the Cornish 2. Fisher Expansion for the F Distribution. Master's Thesis, Texas Tech University, Lubbock, Tx.

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Johnson, N. L., Nixon, E., Amos, D. E. and Pearson, E. S. (1963). "Table of Percentage Points of Pearson Curves for Given /FT and 32 Expressed in Standard Measure." Biometrika, Vol. 50, 459-498.

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255-272.

Lachenbruch, P. A. (1967). The Non-Central F-Distribution; Some Extensions of Tang's Tables. Mimeo Series No. 531, Institute of Statistics, University of North Carolina, Chapel Hill N.C.

Mudholkar, G. S., Chaubey, Y. P. and Lin, C. C. (1976). "Some Approximations for the Noncentral - F Distribution." Technometrics, Vol. 18, No. 3, 351-358.

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83

Patnaik, P. B. (1949). "The Non-Central x^ and F-Distribu-tions and Their Applications." Biometrika, Vol. 36, 202-232.

Pearson, E. S. (1959). "Note on Approximation to the Dis­tribution of Non-Central x^»" Biometrika, Vol. 46, 364.

Fearson, E. S. and Hartley, H. 0. (1972). Biometrika Tables for Statisticians, Vol. 2. Cambridge: Cambridge University Press.

Sankaran, M. (1963) . "Approximations to the Non-Central Chi-Square Distribution." Biometrika, Vol. 50, 199-204.

Satterthwaite, F. E. (1946). "An Approximate Distribution of Estimates of Variance Componets." Biometrics, Vol. 2, 110-114.

Severe, N. C. and Zelen, M. (1960). "Normal Approximation to the Chi-Square and Non-Central F Probability Func­tions." Biometrika, Vol. 47, 411-416.

Tiku, M. L. (1965). "Laguerre Series Forms of Non-Central X^ and F-Distributions." Biometrika, Vol. 52, 415-427.

Tiku, M. L. (1966) . "A Note on Approximating to the Non-Central F-Distribution." Biometrika, Vol. 53, 606-610.

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