On the performance of the nearest proportional to size sampling design

ON THE PERFORMANCE OF THE NEAREST PROPORTIONAL TO SIZE SAMPLING DESIGN

A r u n Kumar A d h i k a r y 1 D e p a r tm e n t o f M a th em atic s

U n i v e r s i t y o f N a ir o b i P . O . Box 3 0 1 9 7 N a i r o b i , Kenya.

Key Words a n d P h r a s e s : F i n i t e p o p u l a t i o n ; G e n e r a l i z e d d iffe re n ce e s t i m a t o r ; N e a r e s t p r o p o r t i o n a l to s i z e s a m p l i n g design: R a t io-c u m - g e n e r a l i z e d d i f f e r e n c e e s t i m a t o r .

ABSTRACT

A s s u m i n g a s u p e r - p o p u l a t i o n model the ex pecte d variance of the g e n e r a l i z e d d i f f e r e n c e e s t i ma t o r ( B a s u , 1 9 7 1 ) b a s e d on the n e a r e s t p r o p o r t i o n a l to s i z e sampling d e s i g n i n t r o d u c e d by G a b l e r ( 1 9 8 7 ) is shown to be less than th a t of the same e s t i m a t o r ba se d on an a r b i t r a r y sampling d e s i g n from w h i c h the former d e s i g n i s r e a l i z e d , f ormer s t r a t e g y i s a l s o shown to f a r e b e t t e r than an

1 O n . l e a v e from I n d i a n S t a t i s t i c a l I n s t i t u t e , C a l c u t t a .

3933

u n b i a s e d r a t i o - c u m - g e n e r a l i z e d d i f f e r e n c e e s t i m a t o r based on the n e a r e s t p r o p o r t i o n a l to s i z e s a m p l i n g d e s i g n in the s e n s e of h a v i n g l e ss e x p e c t e d d e s i g n v a r i a n c e un de r the same m o d e l .

1. INTRODUCTION

C o n s i de r a f i n i t e p o p u l a t i o n U of s i z e N and let y ( i = l , . . . ,N)be the v a l u e s of a v a r i a t e y u n d e r enquiry, our problem i s to e s t i ma t e the the p o p u l a t i o n total

N

Y = £ y. on the b a s i s of a sample of a f i x e d s i z e n drawn i = l

from the p o p u l a t i o n w i t h a p r o b a b i l i t y P o ( s ) .

G a b l e r ( 1 9 8 7 ) has i n t r o d u c e d the n e a r e s t proportional to s i z e sa mpl i ng d e s i g n p ( s ) d e f i n e d as *

P * ( s ) = ( E \ ] p o(s) (1.1)

where A ' s ( i = l , . . . , N ) are a l l p o s i t i v e and a r e given by

where

Do

o o ©

1 "^12 ' 1 I N

o O o

2 1 H 2 ^ 2 N

o o o

H I ^ N 2 ■ • • TIn

* * . T (

= Cti_{l '} ‘ ‘ ’Hn5 ’ Hi

(1.2)

X = ( A^,. . . ,\n ) and

b e i n g the f i r s t order i n c l u s i o n p r o b a b i l i t i e s f o r the units f o r the s a mpl i n g d e s i g n P 0 ( s ) | p * ( s ) Ja n d s b ei n g the s e c o n d order i n c l u s i o n p r o b a b i l i t i e s f o r the p a i r s of units f o r the d e s i g n p ( s ) .

G a b l e r ( 1 9 8 7 ) has a l s o d i s c u s s e d how to r e a l i s e p ( s ) starting from an a r b i t r a r y f i x e d sample s i z e ( n ) d e s i g n p ( s) and has c a l l e d s u c h a d e s i g n a -j ps d e s i g n w h i c h satisf i e s N *£ . = n .

i. =1

N

L e t t 1 = 2 ^ 2 be a g e n e r a l i z e d d i f f e r e n c e

# i = l

’•es Hi

estimator ( B a s u , 1 9 7 1 ) b a s e d on s u c h a n ps d e s i g n f o r some real numbers 9. t = l . . . . , N . known or o t h e r w i s e . Our p u r pos e

l,

here i s to i n v e s t i g a t e w h e t h e r t f a r e s b e t t e r than the same e s t i m a t o r b a s e d on the o r i g i n a l d e s i g n p ( s ) v i z .

N t y.

2 _

O i = i

ie s n,

As the c l a s s i c a l r a t i o e s t i m a t o r i s known to be unbiased u n d e r the M i d z u n o-S e n s a m p l i n g scheme ( M i d z u n o , 1952; S e n , 1 9 5 3 ) and as n * p s d e s i g n i s a p r o c e e d i n g of the Midzuno-Sen s a m p l i n g s c he me , we may c o n s i d e r the f o l l o w i n g ra t i o -cu m-ge n e ra l i z e d d i f f e r e n c e e s t i m a t o r

I

y i - e i i<=s o t =

3 --- * I 6 . .

I \

t «ES

which is a l s o u n b i a s e d u n d e r p * ( s ) .

The m o t i v a t i o n f o r i n t r o d u c i n g the a b ov e ra t i o -c u m-ge n e r a l i z e d d i f f e r e n c e e s t i m a t o r i s e v e n t u a l l y to compare i t s r e l a t i v e e f f i c i e n c y w i t h t h a t of t and t .

1 2

2. A MODEL AND THE RESULTS To compare the r e l a t i v e e f f i c i e n c i e s of the above s t r a t e g i e s we p o s t u l a t e a s u p e r -p o p u l a t i o n model H s p e c i f i e d by

E ( y . ) - e , V (y. ) = E ( y - & ) Z = a 2

m *v V m v m v V x

and C ( V , y .)- Em( y . - e . ) ( y 0 V u*j,

j m v t j j

where s are any p o s i t i v e r e a l numbers V i.

W r i t i n g E * ( V * ) a s an o p e r a t o r f o r the expectation

p p

( v a r i a n c e ) w i t h r e s p e c t to the sa mp l i n g d e s i g n p*, we have

the f o l l o w i n g theorem

Theorem 1 . Under the model M, we have

E V * ( t ) > E V * ( t ) , , n

m p 3 m p 1 ( 2 . 1 )

P r o o f . F o l l o w i n g Godambe and Thompson ( 1 9 7 7 ) , we can write

E m v „* _{m p x} = E *V ( t ) + E . A ( t ) - V ( Y )_{p ttj 1 p m 1 tn}

where A ( t ) = E ( t )-E ( Y ) .

m l m l Tn

Now under the model M, we have A ( t ) = 0 and hence m i

E V * ( tm p

N r i

•>

S i m i l a r l y we may chec k that

m p 3

i S3v

By Ca u chy-Schwarz i n e q u a l i t y i t f o l l o w s th a t

(2.2)

(2. 3)

P ( s )

I

PQ (s)

LS3i i S S

J_ V p°( s ) ^{> _L}

Hi. Hi

( 2 . 4 )

How, E V * ( t )-E V * ( t )

m p 3 m p 1

- - i n . L--' . L <- -H i S3l

1 _

*

Hi

> 0

by u s i n g ( 2 . 4 ) .

Remark 1 . The e q u a l i t y h ol d s when \ X. i s c o n s t a n t f or a l l ISs

s with p ( s ) > 0 i . e when V = - V i w hi ch s a t i s f i e s

N 1 n

) V-n° = 1 and in t h a t c a s e t c o n c i d s s w i t h t .

L l " i 1 3

Let us now c o n s i d e r a s i m p l i f i e d form of the above o *

V"iHi-

2 2 o *

model ( t o be c a l l e d model M ) when ov ~o V n . n . . where <* is a posi ti ve r e a l number.

W r i t i n g E V as an o p e r a t o r f o r e x p e c t a t i o n po <■ poJ

(variance) w i t h r e s p e c t to the s a m p l i n g d e s i g n pqJ we have the f o l l w i n g theorem.

Theorem 2 . Under the model M , we have

E V Proof. We have

E V

m p

,(i J 2 E» v ( * J

- Mv ss 1 ^ III - r 1 )J

( 2 . 5 )

so t h a t

E V f t ] - E V . f t l

m p0 1 2J m p I

N f 'i

= _{/ t} ----_o

U

i=± L Hi M

= 1,1 ” 11 \ ) " ° i = 1

N

b e c a u s e H?]

i =i

N N

= i i - 1 as ] T \ ^

i =1 i =i

N

P * ( s ) - 2+1 i = 1 S5i

= Z ( X N K ( s ) ■ 2Z p * ( s ) + 1

s les s

= Y i I po(s) - zI ( I \ ] po( s > + 1

s i . e s s i- es

S i € S N

Remark 2 . We may note tha t the q u a n t i t y Z n K - - )

the d i r e c t e d d i s t a n c e from the d e s i g n p Q to the d e s i g n p as i n t r o d u c e d by G a b l e r ( 1 9 8 7 ) .

Remark 3 . Here the e q u a l i t y hol d s when ^ V = 1 f o r all 5 i € S

1 *

w i t h p ( s ) > 0 i . e when X. = — V i in w h i c h case i,

o i n

c o i n c i d e s w i t h r e s u l t i n g no d i f f e r e n c e b e t we e n t ±

Under a n o t h e r s i m p l i f i e d v e r s i o n of the above model M

where o i s a

2 2 0

(to be c a l l e d model M ) when a. -a

2 ' i o'" i"®i o

positive r e a l n umber, we have the f o l l o w i n g theorem.

Theorem 3 . Under t h e model M , we have E V *

m p (

2

6

Proof. We have_{h a ve} E V *ft }

m P V. 3J

N r

- 1

■ L - q 2

i = 1 -Hi

N -

2 1

L__ v * i = l Hi

N

=

-Jit:

= 0.

Here a l s o

E V

m p

p0 (s>

S3v l€S

O by u s i n g ( 2 . 4 )

Remark 4 . H e r e a l s o t h e e q u a l i t y h o l d s w h e n

constant f o r a l l s w i t h p ( s ) > 0 i . e . when X. = — V i

o v n

which c a s e th e r e is n o t h i n g to c h oo s e b e t w e e n t 2 and t3

have the same

Remark 5 . Under the model M2 , t^ and t 2

expected d e s i g n - v a r i a n c e i . e . E V * [ t 1 = E V It 1 .

m p (. l j m p Q l Z j

Remark 6 . We may n o t e t h a t u n l i k e Theorem 1, in

Theorem 2 ( T h e o r e m 3 ) , the model v a r i a n c e V ( y . ) is assumed m ^\.

, o * f o *0

to be p r o p o r t i o n a l to V-V V I I I . t Ij j d- h) • D w h i c h is n e a r l y

2 *

P ro po rti on a l to p i( p i ) , p ^ s b e i n g the normed s i z e m e a s u r e s

°f the u n i t s . S i m i l a r a s s u m p t i o n s r e g a r d i n g c/z a re a l s o Available in the l i t e r a t u r e . For e x a m p l e , C a s s e l , S a r n d a l and W r e t m a n ( 1 9 7 6 ) i n v e s t i g a t e d o p t i m a l s t r a t e g i e s f o r

e s t i m a t i n g Y w i t h i n a c l a s s of l i n e a r e s t i m a t o r s under a s u p e r - p o p u l a t i o n model in the se n s e of a t t a i n i n g a lower bound on the mo d e l - e x p e c t e d d e s i g n - v a r i a n c e of an e s t i m a t o r . T h e y f o u n d t h a t the lower bound i s a t t a i n e d by a g e n e r a l i z e d d i f f e r e n c e e s t i m a t o r b a s e d on a s a m p l i n g design w i t h i n c l u s i o n p r o b a b i l i t i e s p r o p o r t i o n a l to known s i z e - m e a s u r e s ( W * s , s a y ) o n l y when the m o d e l - e x p e c t a t i on s and s t a n d a r d d e v i a t i o n s are p r o p o r t i o n a l to W ' s .

ACKNOWLEDGEMENT

The a u t h o r i s g r a t e f u l to the r e f e r e e f o r helpful s u g g e s t i o n s t h a t led to an a p p r e c i a b l e improvement over an e a r l i e r d r a f t .

BI B LI O G RAP P HY

B a s u , D . ( 1 9 7 1 ) . An e s s a y on the l o g i c a l f o u n d a t i o n s of s u r v e y s a m p l i n g . P a r t i in : V . P . Godambe and D . A . S p r o t t e d i t i o n , F o u n d a t i o n s of S t a t i s t i c a l I n f e r e n c e , H o l t , R i n e h a r t and W i n s t o n , T o r o n t o , 2 0 3 - 2 4 2 .

C a s s e l , C . M . , S a r n d a l , C. E. and Wretman, J . H. ( 1 9 7 6 ) . Some r e s u l t s in g e n e r a l i z e d d i f f e r e n c e e s t i m a t i o n and g e n e r a l i z e d regression e s t i m a t i o n f or f i n i t e p o p u l a t i o n s .

B i o m e t r i k a , 6 3 , 6 1 5 - 6 2 0 .

G a b l e r , S . ( 1 9 8 7 ) . The n e a r e s t p r o p o r t i o n a l to size sa mpl i ng d e s i g n . Comm. s t a t i s t . - Theor.

M e t h . , 16, H o . 4 , 1 1 1 7 - 1 1 3 1 .

Godambe, V. P. and Th o m p s o n , M. E. ( 1 9 7 7 ) . Robus t n e a r o p t i m a l e s t i m a t i o n in s u r v e y p r a c t i c e . B u l l . I n t . S t a t i s t . I n s t . , A T , 1 2 3 - 1 4 6 . Midzuno, H. ( 1 9 5 2 ) . On the s a m p l i n g system w i t h

p r o b a b i l i t y p r o p o r t i o n a l to sum of s i z e s . A n n . I n s t . S t a t i s t . Math,, 3 , 9 9 - 1 0 7 .

Sen, A. R . ( 1 9 5 3 ) . On the e a t i m a t i o n of v a r i a n c e in s a m p l i n g w i t h v a r y i n g p r o b a b i l i t i e s . J . I n d . S o c . A g r i . S t a t i s t . , 5, 1 1 9 - 1 2 7 .

Received J u n e 199 0; = e v :s ed A i r ! 1 ! 99 ! .

-ecormended by P. S. R. S. Rao, Upi v e r s ' +y o f R o c h e s te r, Rochest er , NY.