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Optimum stratification for allocationproportional to strata totals for simplerandom sampling schemeSurendar S. Yadava a & Ravindra Singh ba Department of Sociology, Michigan State University, East Lansing,Michigan, 48823, U.S.Ab Visiting Associate Professor, Washington State University, Pullman, 99164,WashingtonPublished online: 27 Jun 2007.

To cite this article: Surendar S. Yadava & Ravindra Singh (1984) Optimum stratification for allocationproportional to strata totals for simple random sampling scheme, Communications in Statistics - Theory andMethods, 13:22, 2793-2806

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COMMUN. STATIST.-THEOR. METH., 1 3 ( 2 2 ) , 2793-2806 ( 1984 )

OPTIMUM STRATIFICATION FOR ALLOCATION PROPORTIONAL TO STRATA TOTALS FOR

SIMPLE RANDOM SAMPLING SCHEME

Surendar S. Yadava Ravindra Singh Department of S o c i a l ogy V i s i t i n g Assoc ia te Pro fessor M ich igan S t a t e U n i v e r s i t y Washington S t a t e U n i v e r s i t y Eas t Lansing, M ich igan 48823 Pul lrnan, Washington 99164 U.S.A.

K e y Words and P h r a s e s : m i n i m a l e q u a t i o n s ; strata b o u n d a r i e s ; l i m i t i n g l o w e r bound ; series e x p a n s i o n s .

ABSTRAC'I

The paper cons iders t h e problem o f f i n d i n g optimum s t r a t a

boundar ies when sample s i z e s t o d i f f e r e n t s t r a t a a r e a l l o c a t e d i n

p r o p o r t i o n t o t h e s t r a t a t o t a l s o f t h e a u x i l i a r y v a r i a b l e . T h i s

v a r i a b l e i s a l s o t r e a t e d as t h e s t r a t i f i c a t i o n v a r i a b l e . Min imal

equa t ions , s o l u t i o n s t o which p r o v i d e t h e optimum boundar ies, have

been ob ta ined . Because o f t h e i m p l i c i t n a t u r e o f these equa t ions

t h e i r e x a c t s o l u t i o n s cannot be ob ta ined . There fo re , methods o f

o b t a i n i n g t h e i r approx imate s o l u t i o n s have been presented. A l i m -

i t i n g express ion f o r t h e v a r i a n c e o f t h e e s t i m a t e o f p o p u l a t i o n

mean, as t h e number o f s t r a t a tend t o become l a r g e , has a l s o been

ob ta ined .

2793

Copyright @ 1984 by Marcel Dekker, Inc.

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2794 YADAVA AND SINGH

INTRODUCTION

L e t t h e p o p u l a t i o n under c o n s i d e r a t i o n be d i v i d e d i n t o L s t r a t a

and a s t r a t i f i e d s imp le random sample o f s i z e n be drawn f rom i t ,

t h e sample s i z e i n t h e h - t h s t r a t u m be ing nh so t h a t tnh=n. If y i s

t h e v a r i a b l e under s tudy, an unbiased e s t i m a t e o f p o p u l a t i o n mean i s

g iven by

Where Wh i s t h e p r o p o r t i o n o f u n i t s i n t h e h - t h s t r a t u m and yh i s

t h e s imp le sample mean based on nh u n i t s drawn f rom t h a t s t ra tum.

I f t h e f i n i t e p o p u l a t i o n c o r r e c t i o n s can be i g n o r e d i n each o f

t h e s t r a t a , t h e va r iance o f t h e e s t i m a t o r yst i s g i v e n b y

The var iance i n (1 .2) depends, i n any p a r t i c u l a r case, on t h e

s t r a t a boundar ies and t h e method o f sample a l l o c a t i o n . The problem

o f de te rmin ing optimum s t r a t a boundar ies on t h e s t u d y v a r i a b l e y

and f o r t h e p r o p o r t i o n a l and Neyman a l l o c a t i o n methods was f i r s t

considered by Dalenius (1950) who ob ta ined t h e minimal equat ions

which gave optimum s t r a t a boundar ies as t h e i r s o l u t i o n s . As these

equa t ions were i m p l i c i t i n n a t u r e t h e i r exac t s o l u t i o n s c o u l d n o t

be ob ta ined . Var ious workers have, t h e r e f o r e , appempted t o f i n d

t h e i r approximate s o l u t i o n s (Dal en ius and Gurney, 1951; Aoyama,

1954; Dalenius and Hodges, 1959; Ekman, 1959a, 1959b, 1960; Durb in,

1959; Mahalanobis, 1952; Singh, 1975a; S e t h i , 1963). As t h e

optimum s t r a t i f i c a t i o n on t h e s tudy v a r i a b l e i s n o t f e a s i b l e i n

p r a c t i c e , t h e problem o f such a s t r a t i f i c a t i o n on t h e a u x i l i a r y

v a r i a b l e was cons idered by Taga (1967), Singh and Sukhatme (1969,

1972, 1973), Singh and Parkash (1975) , Singh (1971, 1975b, 1 9 7 5 ~ )

and S e r f 1 i n g (1968) f o r d i f f e r e n t sample a1 l o c a t i o n methods.

I n s i t u a t i o n s where t h e c o e f f i c i e n t o f v a r i a t i o n does n o t d i f f e r

v e r y much f rom s t r a t u m t o s t ra tum t h e o p t i m a l a l l o c a t i o n o f t h e

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OPTIMUM STRATIFICATION 2795

sample t o d i f f e r e n t s t r a t a reduces t o t h e a l l o c a t i o n p r o p o r t i o n a l t o

s t r a t a t o t a l s i . e . , nh a WnYh, h = l , 2, . . . . , L . Th is s i t u a t i o n i s

aga in l i k e l y t o a r i s e when s t r a t i f i c a t i o n i s r e s o t e d t o f o r opera-

t i o n a l convenience (Murthy, 1967). Hansen, Hurwi t z and Madow (1953)

have demonstrated t h a t t h i s procedure o f sample a1 l o c a t i o n i s a l s o

usefu l i n s i t u a t i o n s where a s l i g h t m o d i f i c a t i o n o f t h e s t r a t a

boundar ies does n o t d i s t u r b a p p r e c i a b l y the s t r a t a sampl i n g var iances

However, i n p r a c t i c e t h e a l l o c a t i o n has t o be i n p r o p o r t i o n t o t h e

s t r a t a t o t a l s o f a s u i t a b l y chosen a u x i l i a r y v a r i a b l e x (which w i l l

a l s o be t r e a t e d as s t r a t i f i c a t i o n v a r i a b l e i n t h i s paper) as t h e

va lues o f t h e e s t i m a t i o n v a r i a b l e y w i l l n o t be a v a i l a b l e . The

p resen t paper cons iders t h e problem o f f i n d i n s optimum s t r a t a bound-

a r i e s (OSB) f o r t h i s a l l o c a t i o n method. Min imal equat ions g i v i n g

t h e OSB have been ob ta ined i n s e c t i o n 2. As usual these equat ions

a r e imp1 i c i t i n n a t u r e and cannot be so lved e x a c t l y , h t e methods o f

o b t a i n i n g approx imate ly optimum s t r a t a boundar ies (AOSB) , have been

d e r i v e d i n s e c t i o n 3. A lso , t h e l i m i t i n g lower bound f o r t h e v a r -

i cance o f t h e e s t i m a t e o f p o p u l a t i o n mean, when t h e number o f s t r a t a

L-, has been o b t a i n e d i n s e c t i o n 4. I t i s observed t h a t u n l i k e t h e

s t r a t i f i c a t i o n on t h e s tudy v a r i a b l e , t h e v a r i a n c e i n t h i s case does

n o t reduce t o zero.

MINIMAL EQUATIONS

When t h e a l l o c a t i o n i n d i f f e r e n t s t r a t a i s p r o p o r t i o n a l t o t h e

s t r a t a t o t a l s f o r t h e a u x i l i a r j v a r i a b l e x, we have

nh = ~ w h u h x / u x ; (h=1,2,. . . ,L), . . . . (2 .1 )

where Wh i s t h e p r o p o r t i o n o f p o p u l a t i o n u n i t s i n t h e h - t h s t ra tum,

phx i s t h e mean f o r x i n t h a t s t r a t u m and px denotes t h e whole

p o p u l a t i o n mean f o r t h e v a r i a b l e x.

The v a r i a n c e

sample a l l o c a t i o n

f o r t h e e s t i m a t e o f p o p u l a t i o n mean under t h e

i n (2 .1 ) reduces f rom (1.2) t o

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2796 YADAVA AND SINGH

We now assume t h a t t h e f i n i t e p o p u l a t i o n under c o n s i d e r a t i o n

can be t r e a t e d as a s imp le random sample f rom an i n f i n i t e super-

p o p u l a t i o n w i t h t h e same c h a r a c t e r i s t i c s . I n t h e super -popu la t ion ,

l e t t h e e s t i m a t i o n v a r i a b l e y and t h e s t r a t i f i c a t i o n v a r i a b l e x be

r e l a t e d as

where c ( x ) i s a r e a l va lued f u n c t i o n o f x and e i s t h e e r r o r term

such t h a t E(e /x ) = 0 and v (e /x ) = + ( x ) > 0 f o r a l l va lues o f x i n

t h e range (a,b) o f x w i t h (b-a) < m . Under t h i s model we, the re -

f o r e , have ( r e f . Singh and Sukhatme, 1969; page 517)

vhy = 'Ihc

and

2 where phc and oh, a r e t h e expected v a l u e and v a r i a n c e f o r t h e

f u n c t i o n c ( x ) i n t h e h - t h s t ra tum, h=1,2,. . . . ,L.

Also, i f f ( x ) denotes t h e marg inal d e n s i t y f u n c t i o n f o r t h e

s t r a t i f i c a t i o n v a r i a b l e x, then we have

and

where ( X ~ - ~ , X ~ ) a r e t h e l o w e r and upper boundar ies f o r t h e h - t h

s t ra tum.

From (2.2) and (2.4) we g e t t h e express ion f o r t h e expected

v a r i a n c e o f t h e e s t i m a t o r & t i n t h e super -popu la t ion , as

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OPTIMUM STR4TIFICATION

For o b t a i n i n g t h e optimum s t r a t a boundar ies (OSB) which c o r r e s -

pond t o t h e minimum of v a r i a n c e i n (2 .6 ) we p a r t i a l l y d i f f e r e n t i a t e

t h e v a r i a n c e f u n c t i o n w i t h r e s p e c t t o xh(h=l,2,. . . . ,L-1) and equate

t h e d e r i v a t i v e s t o zero. T h i s g i v e s us t h e min imal equat ions,

s o l u t i o n s t o which correspond t o t h e OSB.

I n t h e v a r i a n c e express ion of t h e e s t i m a t o r YSt, t h e c o e f f i c i e n t "x - i s c o n s t a n t and, the re fo re , t h e m i n i m i z a t i o n o f V ( G t ) i s n e q u i v a l e n t t o t h e m i n i m i z a t i o n o f

On d i f f e r e n t i a t i n g n (xh) p a r t i a l l y w i t h r e s p e c t t o xh we o b t a i n

aWh aw. Wh q+ (h ) - + Wi $I+ ( i ) 4- = 0 ,.. . (2.7) axh axh

where i = h + l and (h ) = (oh:+uhm ) /vhx.

Now u t i l i z i n g t h e d e f i n i t i o n s o f Wh, uhc, aEc e t c ; i n (2.5) i t

can be e a s i l y v e r i f i e d t h a t we have

aWi w i t h s i m i l a r express ions f o r ( i ) - and W fi)

axh i axh '

Thus f rom (2 .7 ) and (2.8) we g e t t h e min imal equat ions as

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YADAVA AND SINGH

The system of equat ions i n (2 .9 ) i s t h e f u n c t i o n o f s t r a t a

parameters which themselves a r e f u n c t i o n s o f t h e s o l u t i o n s of these

equa t ions , Due t o t h e i m p l i c i t n a t u r e o f these equat ions, i t i s

n o t easy t o f i n d t h e i r exac t s o l u t i o n s . I t becomes, t h e r e f o r e ,

necessary t o o b t a i n approx imate s o l u t i o n s .

APPROXIMATE SOLUTIONS TO THE MINIMAL EQUATIONS

To f i n d t h e approc imate s o l u t i o n s t o t h e minimal equa t ions i n

(2 .9) we s h a l l assume t h e e x i s t e n c e o f v a r i o u s p a r t i a l d e r i v a t i v e s

o f f u n c t i o n s f ( x ) , $ ( x ) and c ( x ) appear ing i n t h i s paper and then

o b t a i n t h e s e r i e s expansions o f t h i s system o f equat ions about t h e

p o i n t xh, t h e common boundary o f t h e h - t h and ( h + l ) - t h s t r a t a . The

expansions f o r t h e r i g h t and l e f t - h a n d s ides o f t h e equa t ion (2 .9 )

a r e o b t a i n e d by u s i n g t h e r e l a t i o n s (3 .1 ) through ( 3 . 5 ) i n Singh and

Sukhatme (1969) . For t h e expansion o f t h e r i g h t - h a n d s i d e abou t t h e

p o i n t xh, (y ,x) i n t h e above r e l a t i o n s ( 3 . 1 ) t o (3 .5 ) i s rep laced

by (xh,xhtl) w h i l e f o r t h e l e f t - h a n d s i d e we r e p l a c e (y ,x) by

(xh-lyxh)' We f i r s t c o n s i d e r t h e expansion o f t h e r i g h t - h a n d s i d e o f (2 .9 ) .

L e t ki = X ~ + ~ - X ~ . Then we have on s i m p l i f i c a t i o n

k: 2 ( c ( x h ) - ) = - c ' 2 + c ' f ' t 2 c ' c " k . + 1 6c12 f f U + l 0 f f ' c ' c t

1 c 3f

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OPTIMUM STRATIFICATION

- f f L , $ ' 3 4 ki +mi ,... ( 3 . 3 )

and

where i n t h e express ions w i t h i n t h e b r a c k e t s on t h e r i g h t - h a n d

s ides o f t h e equa t ions (3 .1 ) t o ( 3 . 4 ) , t h e v a r i o u s f u n c t i o n s and

t h e i r d e r i v a t i v e s a r e eva lua ted a t t h e p o i n t x h '

Now on u s i n g t h e r e l a t i o n s (3 .1 ) t o (3 .4) a long w i t h ( 2 . 9 )

one g e t s on s i m p l i f i c a t i o n t h e s e r i e s expansion o f t h e r i g h t - h a n d

s i d e o f t h e min imal equat ions as

R.H.S. = - x I t ( x h 2 C ' 2 + $ ' x - + ) k . 2 + xh3( f ' c ' 2 + 2 f ' c " ) + x h 2 ( f I $ '

h h 1

and s i m i l a r l y t h e expansion of t h e l e f t - h a n d s i d e o f (2 .9 ) i s seen

t o be

@ h h ) L.H.S. = - 2 2 2 Xh3(f ' c 4 2 f ' c I ' )+xh2( f I,$ '

Xh 1+(xh C ' +I$ 'xh-,$)k h -

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YADAVA AND SINGH 2800

Now l e t

and 2 2 Xh 3 ( f y 2 + 2 f l& ' )+x , ( f ' ( ' + f + " - f c l )-xh(3fm1+f '$)+3f+ B3 =

3x;

There fo re , f rom (3.5) t o (3 .8 ) t h e min imal equat ions (2.9)

can be p u t as

th2 B ~ - B ~ - ~ ~ + o ( ~ [ ) = ki2 B ~ + B ~ . ~ ~ + o ( ~ ~ ~ ) ,

which i s e q u i v a l e n t t o

. . . (3.9)

Now proceeding on t h e l i n e s o f Singh and Sukhatme (1969) i t can

be e a s i l y v e r i f i e d t h a t t h e system o f equat ions (2 .9 ) o r equ iv -

a l e n t l y t h e system (3.9) can a l s o be p u t as

which i s aga in e q u i v a l e n t t o

X 3 2 5 3 ~ 2 ( ~ ) dx ( l + O ( k h ) ) = ! '"*' 3 J B 2 ( ~ ) d x 3 ( l + ~ ( k : ) )

Xh- 1 h

( i = h + l , h = l , 2,. . . ,L-1) ,. . .(3.10)

There fo re , i f we have a l a r g e number o f s t r a t a so t h a t t h e

s t r a t a w i d t h s { k h l a r e smal l and t h e i r h i g h e r powers i n t h e

expansion can be neglected, then t h e systems o f equat ions i n (3 .10)

can be approximated by

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OPTIMJM STRATIFICATION

fxh 3JB2(x) dx = cons tan t (say c ) , ...( 3.11) X h - l

where b

c = la3JB2(x ) dx/L,. . . (3 .12)

I t may be noted here t h a t i n a r r i v i n g a t (3 .11) f rom (3 .10) 5

t h e terms o f o r d e r o(m ) , m = sup. (kh ) , have been neg lec ted on (a,b)

b o t h s ides o f (3 .10) .

For de te rmin ing approx imate s o l u t i o n s t o t h e min imal equat ions

i n ( 2 . 9 ) i n p r a c t i c e , one shou ld e v a l u a t e t h e va lue o f c o n s t a n t c

from (3.12) and then determine x l , t h e upper boundary o f t h e

f i r s t s t ra tum, from t h e r e l a t i o n (3.11) which w i l l now reduce t o

X1 / a 3 J B 2 ( ~ ) dx = c,. . . (3 .13)

Once xl i s determined from (3 .13) , r e l a t i o n (3.11) can be used

a g a i n w i t h h = 2 t o de te rmine x2and so on. T h i s process w i l l

t hen f i n a l l y p r o v i d e us t h e approx imate ly optimum s t r a t a

boundar ies (AOSB) f o r t h e a l l o c a t i o n method b e i n g considered.

Thus, we g e t t h e f o l l o w i n g theorem:

THEOREM 3.1: I f t h e f u n c t i o n

i s bounded and possesses f i r s t two d e r i v a t i v e s f o r a l l x i n (a,b),

then f o r a g i v e n va lue o f L t a k i n g equal i n t e r v a l s on t h e c u m u ~ ? t i v e

o f 3JB2(x) y i e l d s approx imate ly optimum s t r a t a boundar ies.

L I M I T EXPRESSION FOR THE VARIANCE V ( G t )

The express ion f o r t h e v a r i a n c e V(Yst) t h a t we s h a l l o b t a i n

i n t h i s s e c t i o n i s p a r t i c u l a r l y i m p o r t a n t i n approx imate ly

otimum s t r a t i f i c a t i o n on t h e a u x i l i a r y v a r i a b l e . T h i s express ion

g i v e s an i n s i g h t i n t o t h e manner i n which t h e v a r i a n c e o f t h e

e s t i m a t e Fst i s reduced w i t h t h e i n c r e a s e i n t h e number o f s t r a t a

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2802 YADAVA AND SINGH

U n l i k e s t r a t i f i c a t i o n on t h e s tudy v a r i a b l e t h e va r iance v(<,),

i n t h i s case, does n o t tend t o zero as L*. For t h i s purpose we p rove t h e f o l l o w i n g lemma:

LEMMA 4 .1 : - I f (xn,xhtl) a r e t h e boundar ies o f t h e i - t h s t ra tum

and ki = xhtl - xh, then

where B2(x) i s as d e f i n e d i n ( 3 . 7 ) .

PROOF:- Using t h e s e r i e s expansions i n powers o f i n t e r v a l w i d t h

ki f o r Wi, vim and f rom t h e r e l a t i o n s (3 .1 ) t o (3 .5) o f Singh

and Sukhatme (1969) one ge ts on simp1 i f i c a t i o n

S i m i l a r l y t h e express ion f o r t h e second term on t h e l e f t - h a n d

s i d e o f (4 .1 ) i n powers o f i n t e r v a l w i d t h ki can be o b t a i n e d by

making use o f T a y l o r ' s theorem. Thus, we have

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OPTIMUM STRATIFICATION 2 803

A f t e r s u b t r a c t i n g t h e e x p r e s s i o n s f o r v a r i o u s d e r i v a t i v e s

and on s i m p l i f y i n g , one f i n a l l y g e t s

On s u b t r a c t i n g (4 .3 ) f r o m (4 .2) we g e t

k . " + f1m)+3 fm~ 1 + o(ki4) 2fxh 3 D

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2804 YADAVA AND S I N G H

T h i s completes t h e proof of t h e lemma.

When t h e a r e o b t a i n e d f rom t h e Cum ~ ~ ( x ) r u l e , then

we have,

Therefore from Lemma 4 . 1 and r e l a t i o n ( 4 3) . t h e expression

f o r t h e v a r i a n c e V(Yst) can be appro xi mat el^ Pu t as

where

and

u x b 3 6 = ( l a 3JB2(x)dx)

i t can be e a s i l y seen t h a t i n o b t a i n i n g t h e express ion ( 4 a 6 ) 4

for t h e var iance of t h e e s t i m a t o r ht. t h e terms o f order

have been neg lec ted i n t h e second term On t h e r i g h t - h a n d side.

From (4.5) i t can be c l e a r l y observed t h a t t h e v a r i a n c e of the

extimator i n case of optimum s t r a t i f i c a t i o n on t h e a u x i l i a r y

v a r i a b l e x, does n o t reduce t o Zero as La.

T h i s g ives t h e f o l l o w i n g theorum f o r t h e case of sample

a l l o c a t i o n p r o p o t i o n a l t o s t r a t a t o t a l s .

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OPTIMUM STRATIFICATION 2805

THEOREM 4.1:- For t h e AOSB o b t a i n e d form t h e cum 3JB2(x) r u l e , we

have

Tim v(Yst) = y /n, L-xo

where i s as d e f i n e d i n (4 .7 ) .

Th is o b s e r v a t i o n i s c o n t r a r y t o t h e one i n case o f s t r a t i f i c a t i o n

on t h e s t u d y v a r i a b l e where t h e v a r i a n c e V(Yst) tends t o zero as

L* . The r e l a t i o n (4.6) g i v e s t h e e x a c t manner i n which t h e

va r iance v(Yst) w i l l approach y / n as t h e v a l u e o f L i s inc reased .

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Dalenius, T. and M.. Gurney, (1951) . The problem o f optimum s t r a t i f i c d t i o n - 1 1 , Skand. Akt . , 34, 133-148.

De len ius , T. and J.L. Hodges, (1959) . Minimum v a r i a n c e s t r a t i f i c a - t i o n , Jour . Amer. S t a t . Assoc., 54, 88-101.

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--R---Y Mahalanobis, P .C . , (1952) . Some aspects o f t h e des ign o f sample

surveys, Sankhya, 12, 1-17.

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Murthy, M.N., (1967) . Sampling Theory and Methods. C a l c u t t a : S t a t i s t i c a l P u b l i s h i n g S o c i e t y .

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Singh, R., (1975a) . On optimum s t r a t i f i c a t i o n f o r p r o p o r t i o n a l a l l o c a t i o n . Sankhya(c), 37, 109-115.

Singh, R., (1975b) . An a1 t e r n a t i v e method o f s t r a t i f i c a t i o n on t h e a u x i l i a r y v a r i a b l e . Sankhya(c), 37, 100-108.

Singh, R., ( 1 9 7 5 ~ ) . A n o t e on optimum s t r a t i f i c a t i o n i n sampl 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 . Aust . J o u r . S t a t . , 27, 12-21.

Singh, R. and 0. Parkash, (1975) . Optimum s t r a t i f i c a t i o n f o r equal a l l o c a t i o n . Ann. I n s t . S t a t . M a , 27, 273-280.

Singh, R . and B . V . Sukhatme, (1969) . Optimum s t r a t i f i c a t i o n , I n s t . S t a t . Math., 21, 515-528.

Singh, R . and B . V . Sukhatme, (1972) . Optinium s t r a t i f i c a t i o n i n sampling w i t h v a r y i n g p r o b a b i l i t i e s . Ann. I n s t . S t a t . Math., 24, 485-494.

Singh, R. and B.V. Sukhatme, (1973) . Optimum s t r a t i f i c a t i o n w i t h r a t i o and r e g r e s s i o n methods o f e s t i m a t i o n . Ann. I n s t . S t a t . Math., 25, 627-633.

Taga, Y . , (1967). On optimum s t r a t i f i c a t i o n f o r t h e o b j e c t i v e v a r i a b l e based on concomitant v a r i a b l e . Ann. I n s t . S t a t . Math., 19, 101-130.

R e c e i v e d A p r i l , 1 9 8 4 ; R e v i s e d May, 1384 and R e t y p e d A u g u s t , 1 9 8 4 .

Recommended by S. Zacks, S t a t e L J n i v e r s i t y o f New York a t B i n g h a m t o n , NY

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