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Moments of sample momentsDerrick S. Tracy aa Department of Mathematics , University of Windsor , Windsor, Ontario, CanadaPublished online: 27 Jun 2007.

To cite this article: Derrick S. Tracy (1984) Moments of sample moments, Communications in Statistics - Theory andMethods, 13:5, 553-562, DOI: 10.1080/03610928408828700

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COMMUN. STATIST.-THEOR. METH., l 3 ( 5 ) , 553-562 (1984)

MOMENTS OF SAWLE MOMENTS

Der r i ck S. Tracy

Department o f Mathematics U n i v e r s i t y of Windsor

Windsor, On ta r io , Canada

Key Words and P h r a s e s : momen t s ; s a m p l i n g m o m e n t s ; f i n i t e p o p u l a t i o n ; s a m p l i n g w i t h o u t r e p l a c e m e n t ; p a r t i t i o n ; m u l t i - p a r t i t i o n ; power sum; c o m b i n a t o r i a l a p p r o a c h .

ABSTRACT

The problem o f f i n d i n g e x p r e s s i o n s f o r sampling moments of

sample moments h a s been a h i s t o r i c a l l y o l d one . This problem i s

t r e a t e d h e r e , w i t h t h e u s e o f p a r t i t i o n s and m u l t i p a r t i t i o n s , f o r

t h e u n i v a r i a t e a s w e l l a s t h e m u l t i v a r i a t e c a s e . The s y s t e m a t i c

c o m b i n a t o r i a l approach minimizes t h e chance o f o m i t t i n g any

c o n t r i b u t i o n s and making e r r o r s i n t h e i r computation. Component-

w i se i d e n t i f i c a t i o n i s made p o s s i b l e , s o e r r o r s can be l o c a t e d .

From t h e complete s e t o f g e n e r a l moment formulae, s p e c i a l c a s e s

may be ob ta ined by i d e n t i f y i n g i d e n t i c a l v a r i a b l e s .

1. INTRODUCTION

The h i s t o r i c a l problem of moments of sample moments i s

t r e a t e d , u s ing p a r t i t i o n s , s o t h a t each component of a formula i s

i d e n t i f i e d by a p a r t i t i o n , and i t s a l g e b r a i c and numer ica l

c o e f f i c i e n t s a r e determined by a set of r u l e s . The work i s i n the

553

Copyright O 1984 by Marcel Dekker, Inc.

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554 TRACY

s p i r i t of Carver and Dwyer. Impor t an t h i s t o r i c a l developments a r e

d e t a i l e d i n Dwyer (1972), t o whom t h i s a r t i c l e i s ded ica t ed .

U n i v a r i a t e and m u l t i v a r i a t e moments a r e t a c k l e d t o g e t h e r . Simple

random sampl ing w i t h o u t rep lacement from a f i n i t e popu la t ion of

s i z e N i s cons ide red .

2. NOTATION

IT1 ,IT S , .. , For i n t e g e r p we c o n s i d e r p a r t i t i o n P = pl .ps

S 71 s i

w i t h p1 > p2 > ... > p s ( z l ) . and p = 1 p i . n = 1 1 1

i '

With sample o f s i n e n , we l e t e = n / ~ . F u r t h e r , le t

and

Thus, C = e 1 , C2 = el - e 2 , C g = e - 3e2 + 2e3 , 1

. . . . C = e - 7e + 1 2 e 3 - 4 1 2 6e4, We a l s o d e f i n e a symbolic

p roduc t C r e q u i r i n g a d d i t i o n of t h e s u b s c r i p t s . Thus, P4 '

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SAMPLE MOMENTS

We fol low t h e u s u a l n o t a t i o n of denot ing popu la t ion and

sample moments by u and m , assuming t h e popu la t ion mean t o be

0 wi thou t l o s s of g e n e r a l i t y . Popu la t ion and sample means a r e

denoted by y i and m i r e s p e c t i v e l y . I f x denotes t h e

c h a r a c t e r i s t i c under obse rva t ion , and ( ) denotes power sums o r

monomial symmetric f u n c t i o n s , then

S i m i l a r l y , f o r b i v a r i a t e popu la t ions , t h e product moment

and, i n gene ra l f o r m u l t i v a r i a t e p o p u l a t i o n s ,

Sample moments a r e s i m i l a r l y de f ined . Thus,

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When t h e sum o f t h e s u b s c r i p t s i s 1 , we u s e m ' a s t h e n

t h e sample means a r e i m p l i e d , a n d s i m i l a r l y f o r t h e p o p u l a t i o n .

We f o l l o w t h e u s u a l n o t a t i o n i n t h e l i t e r a t u r e f o r moments of

sample moments. Thus,

r 2

E ( m r s tmuvw = E(m rst m rs t m uvw ) = ( s t w 9 . For example ,

i n a t r i v a r i a t e p rob lem.

3. FORMULAE FOR MOMENTS OF MOMENTS

B y p a s s i n g t h e c o m b i n a t o r i c s and t h e a l g e b r a i n v o l v e d , we

p r e s e n t t h e r e s u l t i n g f o rmulae . The f i r s t r e s u l t s imply s t a t e s t h a t

M(1) = D l ( U

which i s t h e same a s

n s i n c e e = -

1 N '

Order 2

Here , we o b t a i n

T h i s s t a t e s t h a t

which i s Theorem 2 . 3 o f Cochran (1977, p . 2 5 ) . Dow

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SAMPLE MOMENTS

When the two v a r i a b l e s a r e i d e n t i c a l , one o b t a i n s

Another formula of o r d e r 2 i s

which i s t h e expected va lue of t h e sample covar iance i n terms of

t h e popu la t ion covar iance . When the v a r i a b l e s a r e i d e n t i c a l , we

ge t the well-known

Order 3

The t h r e e r e s u l t s h e r e a r e

.(;E) = ~ ~ ( 1 1 1 )

These can b e t r a n s l a t e d a s

The cases of two v a r i a b l e s i d e n t i c a l o r a l l v a r i a b l e s i d e n t i c a l

l ead t o f u r t h e r s p e c i a l cases o f t h e above formulae. For example

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Two o f t h e t h r e e b a s i c f o r m u l a e , i n a n o t h e r form, a r e g i v e n by

Nath ( 1 9 6 8 ) .

Order 4

For i n t e g e r s 2 and 3, t h e r e a r e no o t h e r n o n - u n i t p a r t i t i o n s

b e s i d e s t h e i n t e g e r s t h e m s e l v e s . Thus, o n l y one t e r m a p p e a r s i n

t h e moment f o r m u l a e , s i n c e a u n i t p a r t i t i o n c o n t r i b u t e s n o t h i n g

when p o p u l a t i o n means a r e assumed 0 w i t h o u t l o s s o f g e n e r a l i t y .

However, f o r i n t e g e r 4 , t h e non-uni t p a r t i t i o n s a r e 4 and 2 2 .

S i m i l a r l y f o r t h e m u l t i p a r t i t e i n t e g e r 1111. The f o r m u l a e f e a t u r e

c o n t r i b u t i o n s o f s u c h p a r t i t i o n s , a n d t h e c o m p l e t e s e t o f fo rmulae

f o r o r d e r 4 a r e a s be low, where we w r i t e K f o r t h e sum

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SAMPLE MOMENTS 559

The f i r s t and t h e l a s t formulae above, i n d i f f e r e n t forms, a r e

g iven by Nath (1963) and by Dwyer and Tracy (1980) . For t h e

b i v a r i a t e c a s e , t he f i r s t formula y i e l d s t h e formulae o f Sukhatme - - - - 2 - - 2

and Sukhatme (1970, p.192) f o r E(y-Y) ( x - x ) ~ and E()-Y) (x-X) . Thus, f o r example,

The u n i v a r i a t e formula f o r ~ ( x - 3 ~ can s i m i l a r l y be ob ta ined .

Order 5

Here t h e i n t e g e r 5 has 5 and 32 a s i t s non-unit p a r t i t i o n s .

For t h e u n i v a r i a t e c a s e , t h e n , one would have

For t h e m u l t i v a r i a t e c a s e , t h e lo(;) would be r ep l aced by t h e sum

11100 00111 of t e n terms [ ) + .. . + ( 1 , which we denote by I .

00011 11000

Thus,

These formulae , i n d i f f e r e n t forms, a r e g iven i n Raghunandanan and

S r i n i v a s a n (1973) and Dwyer and Tracy (1980) . Dow

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The f i v e o t h e r formulae , comple t ing t h e s e t o f a l l moment

formulae o f o r d e r 5 , a r e

= (D3-3D4+2D5)(l l l l l ) + (D21-3D22+2D32)

10110 01110 11001 (2D32-D22) ( ( b ) + (01001) + (100011 + (001101

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SAMPLE MOMENTS

Order 6

The i n t e g e r 6 h a s f o u r non-uni t p a r t i t i o n s , i . e . , 6 , 42, 33,

222. The two s t r a i g h t f o n v a r d fo rmu lae h e r e a r e

I n t h e m u l t i v a r i a t e c a s e , t h e s e v e r a l power sums have t o b e s p e l t

o u t . Thus, 151;) would change t o t h e sum

(111100] 000011 + . . , + (yyiiii) , c o n s t i t u t i n g 1 5 t e r n s , deno t ed below

b y R . S i m i l a r l y s e t t i n g

110000

we have

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TRACY

4. CONCLUSION

Using a combinator ia l approach, a l l p o s s i b l e moments of

moments formulae f o r any o r d e r can be w r i t t e n out f o r t h e most

genera l m u l t i v a r i a t e case . Spec ia l cases , when some v a r i a b l e s a r e

i d e n t i c a l , can then be ob ta ined by coa lesc ing .

Cochran, W . G . , (1977). S a m p l i n g T e c h n i q u e s . New York: John Wiley & Sons.

Dwyer, P. S . , (1972). Moment func t ions of sample moment func t ions . S y m m e t r i c F u n c t i o n s i n S t a t i s t i c s , ed. D . S. Tracy, Unive r s i ty of Windsor, Windsor, 11-51.

Dwyer, P. S. and Tracy, D. S . , (1980). Expecta t ion and e s t i m a t i o n of product moments i n sampling from a f i n i t e popula t ion. J . Amer . S t a t i s t . A s s o c . , 7 5 , 431-437.

Nath, S . N . , (1968). On product moments from a f i n i t e un ive r se . J . Amer . S t a t i s t . A s s o c . , 6 3 , 535-541.

Raghunandanan, K. and Sr in ivasan , R . , (19 73) . Some product moments u s e f u l i n sampling theory. J . Amer . S t a t i s t . A s s o c . , 6 8 , 409-413.

Sukhatme, P . V. and Sukhatme, B . V . , (1970) . S a m p l i n g T h e o r y of S u r v e y s w i t h A p p l i c a t i o n s . Iowa S t a t e Univ. P ress , Ames, Iowa.

R e c e i v e d b y E d i t o r i a l Board member J u n e , 1 9 8 3 ; R e v i s e d O c t o b e r , 1 9 8 3 .

Recommended b y A . M . M a t h a i , M c G i l l U n i v e r s i t y , M o n t r e a l , Canada

R e f e r e e d b y J . C. K o o p , 3201 C l a r k A v e . R a l e i g h , NC

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