This article was downloaded by: [Dalhousie University]On: 07 October 2014, At: 14:50Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in Statistics - Theory and MethodsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsta20

AN IMPROVED LOWER BOUND FOR LEHMANN'SLAMBDAJohn D. Spurrier aa Department of Statistics , University of South Carolina , Columbia, 29208, SouthCarolinaPublished online: 02 Sep 2006.

To cite this article: John D. Spurrier (2002) AN IMPROVED LOWER BOUND FOR LEHMANN'S LAMBDA, Communications inStatistics - Theory and Methods, 31:2, 193-199, DOI: 10.1081/STA-120002645

To link to this article: http://dx.doi.org/10.1081/STA-120002645

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

INFERENCE

AN IMPROVED LOWER BOUND

FOR LEHMANN’S LAMBDA

John D. Spurrier

Department of Statistics, University of South Carolina,Columbia, South Carolina 29208

ABSTRACT

Let X1, . . .,X7 be i.i.d. random variables with a commoncontinuous distribution F. The parameter �(F )¼P(X1þ

X4<X2þX3 and X1þX7<X5þX6), which appears in themoments of some rank statistics, has been studied by severalauthors. It is shown that the existing lower bound for �(F ) of0.28254 can be improved to 0.28452. The new lower bound is22.6% closer to the existing upper bound of 0.29132.

Key Words: Order statistics; Nonparametric; Probabilitybound

1. INTRODUCTION

Let X1, . . . ,Xn be n i.i.d. random variables with a common continuousdistribution F where n� 7. Define �(F )¼P(X1þX4<X2þX3 and X1þ

X7<X5þX6). Lehmann (1), Hollander (2,3), Doksum (4), Hsu (5), andSpurrier (6) give examples in which the moments and limiting distributionsof rank statistics depend on �(F ).

193

Copyright & 2002 by Marcel Dekker, Inc. www.dekker.com

COMMUN. STATIST.—THEORY METH., 31(2), 193–199 (2002)

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

Although �(F ) depends on F, the interval of possible values is verynarrow. Lehmann (1), Hollander (3), Mann and Pirie (7), and Spurrier (8,9)have derived bounds for �(F ). Spurrier (8) gives the best lower bound,�(F )� 89/315� 0.28254. Spurrier (9) gives the best upper bound,

�ðFÞ � f7 ½1 ð2=3Þ1=2�2=4g=24 � 0:29132:

It is shown in the next section that the lower bound can be improved to�(F )� 0.28452. This improved bound reduces the interval of possible �(F )values by 22.6%.

2. DERIVATION OF NEW LOWER BOUND

Let G(a, b, c, d, e, f, g) be the indicator function for the event

fa þ d < b þ c and a þ g < e þ f g: ð1Þ

Note that G is invariant to interchanges of its second and third argumentsand interchanges of its fifth and sixth arguments. Arguments d and g have aparallel nature. Now,

E [G(X1, . . . ,X7)]¼ �(F ).

Let X(1)<X(2)< <X(n) denote the order statistics of X1, . . . ,Xn. LetY(1)<Y(2)< <Y(7) denote the order statistics of an arbitrary subsetof seven X’s. Spurrier (8) achieves the existing lower bound for �(F ) byshowing that the unbiased estimator

�̂� ¼X

GðYi1 , . . . ,Yi7 Þ=630 � 178=630 ð2Þ

where the summation is over the 630 permutations (i1, . . . , i7) of (1, 2, 3, 4,5, 6, 7) having i2<i3, i5<i6, and i4<i7. He showed that �̂� can be determinedfrom indicator functions for a series of inequalities involving Y(1), . . . ,Y(7).Using a computer search of all achievable combinations of these indicatorfunction values, he found that at least 178 summands in the numerator of �̂�must equal one. Moreover, the numerator equals 178 if and only if allinequalities in columns A and B or in columns A and C of Table 1 aretrue. For the numerator to equal 179, all inequalities in column A exceptA4 and A7 must be true and one of 17 other sets of inequalities must be true.If the numerator does not equal 178 or 179, it is at least 180.

One can improve the lower bound for �(F ) by taking n>7 andaveraging �̂� over all subsamples of seven X’s. For example, let n¼ 8 and

194 SPURRIER

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

let �̂�i denote the value of �̂� for the subset of (X1, X2, . . . ,X8) obtained byremoving the ith order statistic, X(i), i¼ 1, . . . , 8. Now, �̂�½8� ¼ �i �̂�i=8 is anunbiased estimator of �(F ). Let Ni denote the numerator of �̂�i, i¼ 1, . . . , 8.The following theorem provides an improved lower bound for �(F ) bybounding �̂�[8].

Theorem 1. �̂�[8] and �(F )� 717/2520¼ 0.28452.

Proof. The proof involves four types of (Ni,Nj) pairs. The first type of pairis such that Ni and Nj cannot simultaneously be�179. Consider (N1,N3).The numerator N1 is computed on the ordered sample

Yð1Þ ¼ Xð2Þ, Yð2Þ ¼ Xð3Þ, Yð3Þ ¼ Xð4Þ, Yð4Þ ¼ Xð5Þ,

Yð5Þ ¼ Xð6Þ, Yð6Þ ¼ Xð7Þ, Yð7Þ ¼ Xð8Þ: ð3Þ

The numerator N3 is computed on the ordered sample

Yð1Þ ¼ Xð1Þ, Yð2Þ ¼ Xð2Þ, Yð3Þ ¼ Xð4Þ, Yð4Þ ¼ Xð5Þ,

Yð5Þ ¼ Xð6Þ, Yð6Þ ¼ Xð7Þ, Yð7Þ ¼ Xð8Þ: ð4Þ

Table 1. Necessary and Sufficient Conditions for Numerator of �̂� to Equal 1781 andNecessary Conditions for the Numerator to Equal 1792

Column A Column B Column C

A1: Y(1)þY(4)<Y(2)þY(3) B1: Y(1)þY(7)<Y(3)þY(5) C1: Y(1)þY(5)<Y(2)þY(3)

A2: Y(1)þY(6)<Y(2)þY(5) B2: Y(2)þY(4)<Y(1)þY(5) C2: Y(1)þY(6)<Y(2)þY(4)

A3: Y(1)þY(6)<Y(3)þY(4) B3: Y(2)þY(6)<Y(1)þY(7) C3: Y(1)þY(7)<Y(2)þY(6)

A4: Y(1)þY(7)<Y(3)þY(6) B4: Y(2)þY(6)<Y(3)þY(5) C4: Y(2)þY(7)<Y(4)þY(6)

A5: Y(1)þY(7)<Y(4)þY(5) B5: Y(3)þY(4)<Y(2)þY(5) C5: Y(3)þY(6)<Y(4)þY(5)

A6: Y(2)þY(3)<Y(1)þY(6) B6: Y(4)þY(6)<Y(2)þY(7) C6: Y(3)þY(5)<Y(1)þY(7)

A7: Y(2)þY(5)<Y(1)þY(7) B7: Y(5)þY(6)<Y(3)þY(7) C7: Y(3)þY(5)<Y(2)þY(6)

A8: Y(2)þY(6)<Y(4)þY(5) C8: Y(3)þY(7)<Y(4)þY(6)

A9: Y(2)þY(7)<Y(5)þY(6)

A10: Y(3)þY(4)<Y(1)þY(7)

A11: Y(3)þY(4)<Y(2)þY(6)

A12: Y(3)þY(6)<Y(2)þY(7)

A13: Y(4)þY(5)<Y(2)þY(7)

A14: Y(5)þY(6)<Y(4) þ Y(7)

1Numerator¼ 178 if and only if all inequalities in columns A and B or in columns A

and C hold.2If numerator¼ 179 then all inequalities in column A except A4 and A7 must hold.

LOWER BOUND FOR LEHMANN’S LAMBDA 195

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

From Table 1, one of the necessary inequalities for N1� 179 is A3:Y(1)þY(6)<Y(3)þY(4). In terms of the X’s, we have

Xð2Þ þ Xð7Þ < Xð4Þ þ Xð5Þ: ð5Þ

One of the necessary inequalities for N3� 179 is A11: Y(3)þY(4)<Y(2)þY(6). In terms of the X’s, we have

Xð4Þ þ Xð5Þ < Xð2Þ þ Xð7Þ: ð6Þ

Conditions (5) and (6) are contradictory. Thus, N1 and N3 cannot bothbe �179.

Similar arguments involving the inequalities in column A of Table 1yield that other pairs of Ni and Nj cannot simultaneously be �179. Thesepairs are denoted by the notation (Ap,Aq) in the (Ni,Nj) cell of Table 2,where ‘‘Ap’’ is an inequality that must hold for Ni� 179 and ‘‘Aq’’ is aconflicting inequality that must hold for Nj� 179. Based on the argumentin the previous paragraph, we have (A3,A11) for the (N1,N3) cell.

The second type of (Ni,Nj) pairs are such that it is impossible tosimultaneously have Ni� 179 and Nj¼ 178. Consider (N1,N6). Fromcolumn A of Table 1, two of the necessary inequalities for N1� 179 are

A1: Yð1Þ þ Yð4Þ < Yð2Þ þ Yð3Þ and A11: Yð3Þ þ Yð4Þ < Yð2Þ þ Yð6Þ: ð7Þ

In terms of the X’s, we have

Xð2Þ þ Xð5Þ < Xð3Þ þ Xð4Þ and Xð4Þ þ Xð5Þ < Xð3Þ þ Xð7Þ: ð8Þ

From columns B and C of Table 1, the necessary inequalities for N6 to equal178 must include either

B5: Yð3Þ þYð4Þ < Yð2Þ þYð5Þ or C5: Yð3Þ þYð6Þ < Yð4Þ þYð5Þ: ð9Þ

In terms of the X’s, we have either

Xð3Þ þ Xð4Þ < Xð2Þ þ Xð5Þ or Xð3Þ þ Xð7Þ < Xð4Þ þ Xð5Þ: ð10Þ

Conditions (8) and (10) are contradictory. Thus, we cannot have N1� 179and N6¼ 178.

Similar arguments involving the inequalities in columns A, B, and C ofTable 1 hold for the pairs (N1,N7), (N8,N2), (N8,N3), and (N8,N4). The twopairs of inequality contradictions yielding that we cannot simultaneouslyhave Ni� 179 and Nj¼ 178 are indicated by (Ap, Bq) and (Ar, Cs) in the(Ni,Nj) cell of Table 2.

196 SPURRIER

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

Table

2.

ContradictionsInvolving(N

i,N

j)Pairs1

N1

N2

N3

N4

N5

N6

N7

N8

N1

(A3,A

11)

(A5,A

13)

(B1,A

13)

(C3,A

12)

(A1,B

5)

(A11,C

5)

(A1,B

5)

(A12,C

8)

(A3,A

13)

N2

(A6,A

3)

(A6,A

3)

(A10,A

5)

(B2,A

3)

(C5,A

14)

N3

(A11,A

3)

(A10,A

5)

(A10,A

5)

(B2,A

2)

(C5,A

14)

N4

(A13,A

5)

(A13,A

9)

(B4,A

12)

(C1,A

6)

N5

(B7,A

9)

(C7,A

2)

(A3,A

6)

(A3,A

10)

N6

(B5,A

1)

(C5,A

11)

(A3,A

6)

(A5,A

10)

(A3,A

10)

N7

(B5,A

1)

(C8,A

12)

(A5,A

10)

(A5,A

10)

(A9,A

13)

N8

(A13,A

3)

(A3,B

2)

(A14,C

5)

(A2,B

2)

(A14,C

5)

(B3,A

2)

(C6,A

3)

(A10,A

3)

(A10,A

3)

1Thenotation

(Ap,A

q)in

the(N

i,N

j)cell

implies

thatinequality

Ap

from

Table

1forthesample

without

Xicontradicts

inequalityAq

from

Table

1forthesample

without

Xj.

Hence

Niand

Njcannotsimultaneously

be�179.The

notation

(Ap,B

q),

(Ar,Cs)

inthe(N

i,N

j)cellim

plies

that

Ni�179and

Nj¼178are

contradictory.Thenotation(Bp,A

q),

(Cr,As)

inthe(N

i,N

j)cellim

plies

that

Ni¼178and

Nj�179are

contradictory.

LOWER BOUND FOR LEHMANN’S LAMBDA 197

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

A third type of (Ni,Nj) pair is such that we cannot have Ni¼ 178and Nj� 179. This is the dual of the second type. The pairs of inequalitycontradictions for this type (Ni,Nj) pair are indicated by (Bp,Aq) and(Cr,As) in the (Ni,Nj) cell of Table 2.

The fourth type of (Ni,Nj) pair has no restrictions on values of Ni

and Nj. Such pairs have no entry in the (Ni,Nj) cell of Table 2.A computer search of all (N1, . . . ,N8) values consistent with the restric-

tions from Table 2 yields that �̂�[8] must be at least 179.25/630. There are four(N1, . . . ,N8) values yielding �̂�[8]¼ 179.25/630. One of these has N2¼N3¼

N4¼ 178 and the other Ni’s¼ 180. As �̂�[8] is an unbiased estimator of �(F ),we have �(F )� 179.25/630¼ 717/2520. The proof is complete.

Similar arguments can be used for n>8, but the numbers of numera-tors and possible contradictions becomes unwieldy. Based on some limitedcomputer searches for n¼ 9 and 10, the author conjectures that argumentsanalogous to the proof of Theorem 1 would yield lower bounds for �(F )approaching 180/630¼ 2/7¼ 0.28571 as n ! 1:

ACKNOWLEDGMENT

The author thanks the referee for helpful comments.

REFERENCES

1. Lehmann, E.L. Asymptotically Nonparametric Inference in SomeLinear Models with One Observation Per Cell. Annals of MathematicalStatistics 1964, 35 (2), 726–734.

2. Hollander, M. An Asymptotically Distribution-Free Multiple Compar-isons Procedure—Treatments vs. Control. Annals of MathematicalStatistics 1966, 37 (3), 735–738.

3. Hollander, M. Rank Tests for Randomized Blocks When the Alterna-tives have an a Priori Ordering. Annals of Mathematical Statistics 1967,38(3), 867–877.

4. Doksum, K.A. Robust Procedures for Some Linear Models with OneObservation Per Cell. Annals of Mathematical Statistics 1967, 38(3),878–883.

5. Hsu, J.C. Simultaneous Inference with Respect to the Best Treatmentsin Block Designs. Journal of the American Statistical Association 1982,77(378), 461–477.

6. Spurrier, J.D. Distribution-Free and Asymptotically Distribution-FreeComparisons with a Control in Blocked Experiments. Multiple

198 SPURRIER

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

ORDER REPRINTS

Comparisons, Selection, and Applications in Biometry: A Festschrift inHonor of Charles W. Dunnett, F.M. Hoppe, Ed.; Marcel Dekker:New York 1993, 97–119.

7. Mann, B.L.; Pirie, W.R. Tighter Bounds and Simplified Estimation forMoments of Some Rank Statistics. Communications in Statistics—Theory and Methods 1982, 11, 1107–1117.

8. Spurrier, J.D. Improved Bounds for Moments of Some Rank Statistics.Communications in Statistics—Theory and Methods 1991, 20(8),2603–2608.

9. Spurrier, J.D. More Bounds for Moments of Some Rank Statistics.Communications in Statistics—Theory and Methods 1995, 24(10),2679–2682.

LOWER BOUND FOR LEHMANN’S LAMBDA 199

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014

Order now!

Reprints of this article can also be ordered at

http://www.dekker.com/servlet/product/DOI/101081STA120002645

Request Permission or Order Reprints Instantly!

Interested in copying and sharing this article? In most cases, U.S. Copyright Law requires that you get permission from the article’s rightsholder before using copyrighted content.

All information and materials found in this article, including but not limited to text, trademarks, patents, logos, graphics and images (the "Materials"), are the copyrighted works and other forms of intellectual property of Marcel Dekker, Inc., or its licensors. All rights not expressly granted are reserved.

Get permission to lawfully reproduce and distribute the Materials or order reprints quickly and painlessly. Simply click on the "Request Permission/Reprints Here" link below and follow the instructions. Visit the U.S. Copyright Office for information on Fair Use limitations of U.S. copyright law. Please refer to The Association of American Publishers’ (AAP) website for guidelines on Fair Use in the Classroom.

The Materials are for your personal use only and cannot be reformatted, reposted, resold or distributed by electronic means or otherwise without permission from Marcel Dekker, Inc. Marcel Dekker, Inc. grants you the limited right to display the Materials only on your personal computer or personal wireless device, and to copy and download single copies of such Materials provided that any copyright, trademark or other notice appearing on such Materials is also retained by, displayed, copied or downloaded as part of the Materials and is not removed or obscured, and provided you do not edit, modify, alter or enhance the Materials. Please refer to our Website User Agreement for more details.

Dow

nloa

ded

by [

Dal

hous

ie U

nive

rsity

] at

14:

50 0

7 O

ctob

er 2

014