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Estimation of mean using double samplingfor stratification and multivariate auxiliaryinformationT. P. Tripathi a & Shashi Bahl ba Stat-Math. Division, Indian Statistical Institute, Calcutta, 700 035,Indiab Dept. of Mathematics, M. D. University, Rohtak, 124 001, IndiaPublished online: 27 Jun 2007.

To cite this article: T. P. Tripathi & Shashi Bahl (1991) Estimation of mean using double sampling forstratification and multivariate auxiliary information, Communications in Statistics - Theory and Methods,20:8, 2589-2602, DOI: 10.1080/03610929108830652

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COMMUN. STATIST.-THEORY METH., 2 0 ( 8 ) , 2589-2602 ( 1 9 9 1 )

ESTIMATION OF MEM! USING DOUBLE SAMPLING FOR STRATIFICATION AND MULTIVARIATE AUXILIARY

To P o T r i p a t h i

S ta t -Math . D i v i s i o n , 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 - 700 035, I n d i a

S h a s h i Bahl

Dept. o f Mathemat ics , M. D. U n i v e r s i t y , 20htak - 124 031, I n d i a

Key i'irords and P h r a s e s : Combined and S e p a r a t e E s t i m a t o r s ; R e l a t i v e Per formance , S t r a t i - f i c a t i o n .

ABSTRACT

S e v e r a l e s t i m a t o r s f o r e s t i m a t i n g t h e mean o f

a p r i n c i p a l v a r i a b l e a r e p roposed based on d o u b l e

s ampl ing f o r s t r a t i f i c a t i o n (DSS) and m u l t i v a r i a t e

a u x i l i a r y i n f o r m a t i o n . The g e n e r a l p r o p e r t i e s o f

t h e p roposed e s t i m a t o r s a r e s t u d i e d , s e a r c h f o r

optimum e s t i m a t o r s i s made and t h e p roposed e s t i -

ma to r s a r e compared w i t h t h e c o r r e s p o n d i n g e s t ima-

t o r s b a s e d on u n s t r a t i f i e d d o u b l e s ampl ing (USDS).

1. INTRODUCTION

When t h e s ampl ing f r a m e w i t h i n s t r a t a i s

known, s t r a t i f i e d s ampl ing i s used ; b u t t h e r e a r e

Copyright O 1991 by Marcel Dekker, Inc.

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2590 T R I P A T H I AND B A H L

many s i t u a t i o n s o f p r a c t i c a l i m p o r t a n c e w h e r e t h e

s t r a t a w e i g h t s a r e known and t h e f r a m e w i t h i n s t r a -

t a i s n o t a v a i l a b l e . I n t h e s e s i t u a t i o n s t h e t e c h -

n i q u e of p o s t - s t r a t i f i c a t i o n may b e employed ,

However i n o t h e r s i t u a t i o n s s t r a t a w e i g h t s may n o t

b e known e x a c t l y a s t h e y become o u t d a t e d w i t h t h e

p a s s a g e o f t i m e . F u r t h e r t h e i n f o r m a t i o n o n s t r a -

t i f i c a t i o n v a r i a b l e may n o t b e r e a d i l y a v a i l a b l e

b u t c o u l d b e made a v a i l a b l e by d i v e r t i n g a p a r t o f

t h e s u r v e y b u d g e t . Under t h e s e c i r c u m s t a n c e s t h e

method o f d o u b l e s a m p l i n g f o r s t r a t i f i c a t i o n (DSS)

c a n b e used .

I n t h e p r o p o s e d DSS Scheme we s e l e c t a p r e l i -

m i n a r y l a r g e s a m p l e S o f s i z e n t r a t h e r i n e x p e n - ( 1 )

s i v e l y f r o m a p o p u l a t i o n o f N u n i t s w i t h s i m p l e

random 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 (SRS'VVO3) a n d

o b s e r v e t h e a u x i l i a r y v a r i a b l e s x l ,x2 , . . . ,X Let P '

( x i j ) , i = 1 ,2 , . . . , p ; j = 1 , 2 , . . . , n 1 d e n o t e t h e

- ,n x - o b s e r v a t i o n s a n d x1 - Z x . . / n t , t h e s a m p l e

i - j=l 1 J

means. The s a m p l e S i s t h e n s t r a t i f i e d i n t o L ( 1 )

s t r a t a on t h e b a s i s o f i n f o r m a t i o n f o r o n e o r more

x i ' s o b t a i n e d t h r o u g h S ( 1 ) '

Let n; d e n o t e t h e num-

b e r o f u n i t s i n S f a l l i n g i n t o h - th s t r a t u m ( 1 )

( h = 1 , 2 ,,.., L, C nA = n l ) y i e l d i n g t h e r e p r e s e n t a - h

t i o n n - - - h n

h x i = ;wI; x i h w h e r e x i h = E xijh/nl; and w1 = - j=1 ti n '

S u b s a m p l e s o f s i z e s nh = vhn; 0 < v h < 1; h=1 ,2 , . . ,L ,

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

i n d e p e n d e n t l y , u s i n g SRS'iVOR w i t h i n e a c h s t r a t u m a n d

y , t h e v a r i a b l e o f main i n t e r e s t i s o b s e r v e d , L e t

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ESTIMATION OF MEAN USING DOUBLE SAMPLING 2591

n = Z nh. and ( y j h ) , j = 1,2 ?.... n . h = l hPnh

h = 1 , 2 , . . . , L d e n o t e y -obse rva t ions and y = 1 y /nh h j=l j h

C l e a r l y wA i s an u n b i a s e d e s t i m a t o r o f s t r a t a we igh t s h l

- S i m i l a r l y t h e sample means and xidS based on

f i r s t sample and subsample r e s p e c t i v e l y a r e unb ia sed - e s t i m a t o r s o f p o p u l a t i o n mean Xi = .XwhFih of a u x i l i a r y

h v a r i a b l e x ; . For e s t i m a t i n g t h e p o p u l a t i o n mean F, t h e

I

cus tomary u n b i a s e d e s t i m a t o r based on DSS and i t s

v a r i a n c e a r e g i v e n by - - Yds = c w ' Y

h h (1 .1)

Some e s t i m a t o r s b a s e d on DSS and i n f o r m a t i o n on a

s i n g l e a u x i l i a r y v a r i a b l e have been proposed by I g e

and T r i p a t h i (1987) f o r improving t h e p r e c i s i o n o f

e s t i m a t i o n compared t o Gs. I n t h i s p a p e r we d i s c u s s

s e v e r a l methods o f e s t i m a t i o n , b a s e d on m u l t i v a r i a t e

a u x i l i a r y i n f o r m a t i o n , a s a n e f f o r t f o r f u r t h e r i m -

provement o f p r e c i s i o n o f e s t i m a t i o n .

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2592 TRIPATHI AND BAHL

2. MULTIVARIATE COMBINED AND SEPARATE ESTIMATORS BASED ON DSS

U t i l i z i n g t h e i n f o r m a t i o n c o l l e c t e d on x - v a r i a t e s

t h r o u g h t h e p r e l i m i n a r y s a m p l e S (1 ), we d e f i n e m u l t i - . . v a r i a t e combined d i f f e r e n c e , r a t l o a n d ra t io -cum-

p r o d u c t e s t i m a t o r s i n DSS by

P e = 1 a ia i

i=l - - - e = e

DMC if ai= yds-Ai(xids- x!) i = 1 , 2 , . . , p ( 2 . 1 )

1

e = e - yds - RPL4c if ai - 7 x t f o r i=l, . . . , q i ( 2 . 3 ) X i d s

P w h e r e a = ( a l , a 2 , . . . ,

) ' w i t h E a . = 1 is a weigh-

i=l 1

f u n c t i o n and Aiis a r e s u i t a b l y c h o s e n c o n s t a n t s .

U s i n g t h e same amount o f i n f o r m a t i o n , we c a n de-

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

r a t i o - c u m - p r o d u c t e s t i m a t o r s i n DSS b y

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E S T I M A T I O N O F MEAN U S I N G D O U B L E S A M P L I N G

The v a r i a b l e s x1,x2, ..., x (2 i n e RPMC and e~~~~

a r e t h o s e o n e s o f x which a r e p o s i t i v e l y c o r r e l a t e d

w i t h y. o b v i o u s l y eDMC and e a r e u n b i a s e d f o r P E r n and e x a c t e x p r e s s i o n s f o r t h e i r v a r i a n c e s a r e g i v e n by

- - w i t h Bh - (bh ik) : Dh - ( d h i k ) i , k = l , . . . , p

- 2 bhik - Sho-h i S h o i - 'kShokf ' ihkShik

2 dhik = S h o - h ~ h S h o i - A kh S h o k f h i h A kh S h i k

hTh - - z ( x ~ ~ ~ - X ~ ~ ) ( X ~ ~ ~ - ~ ~ ~ ) i , k = O , l , *.,P where Shik=

h j=l

t h e s u b s c r i p t s 0 , 1 , 2 , . . . , p r e f e r i n g t o t h e v a r i a b l e s

r e s p e c t i v e l y . P F o r l a r g e s a m p l e s , t h e a p p r o x i m a t e e x p r e s s i o n s

f o r t h e b i a s e s and MSES o f t h e e s t i m a t o r s e R.FK ' R:.iS '

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2594 TRIPATHI AND BAHL

7 h4(eRdC) = V(eDMC) w i t h Ai = R. = - (2 .9 )

F:

i h I h (2 .10) M(eHvIS) = V(eDMS) w i t h Aih = R = =-- 'ih

ivi(eRpMC)= V(eDh4c) w i t h A. = H. A = R, 1 1' k ..

for e a c h i , k = 1 , 2 , . . , q

A . = -Ri, A = -Rk 1 k

for each i , k = q + l , . . , p

A. = Fii , Ak= -Rk 1

f o r i = 1 ,2 , . . , , q ; k = q + l , ...,p.

(2.11)

M(eRFMS)= V(eDh!S) w i t h Aih = Rib; A k h = Rkh

f o r e a c h i , k = l , 2 , .. , q - Aih - -R ih ;Akh = -Rkh

f o r e a c h i, k=q+l, . . ,p

h ih - - Rib; A k h = -Rkh

f o r i = l,..o,q,

k = q + l , . . , p .

(2 .12)

Using t h e r e s u l t s of Rao (1973) , non-negat ive

unb ia sed e s t i m a t o r s o f V(eDMC) and V(eDhlS) a r e g i v e n

b Y 1-f s2 1 1 2

v(eDMC) = 7 + ;it'(; - l ) w i ( ' ~ a . a (she-hishoi h h i k 1 k

-Akshok+A A s ) i k h i k

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E S T I M A T I O N O F MEAN U S I N G D O U B L E S A M P L I N G 2595

a n d

2 w i t h she = S h o o

F u r t h e r , n o n - n e g a t i v e b u t b i a s e d e s t i m a t o r s f o r

t h e MSES o f eaIIC, e e e RMS' 9PMC' RPMS a r e g i v e n by

rn(emG) = v(eDbtC) w i t h A. 1 = ri = Yds / 'ids

- m(eRpMC)= v(eDbIc) w i t h hi - r.i; A k = rk

f o r e a c h i , k = 1 , 2 , . . . , q - Ai - -ri; A k = -r k

rn(ewMS) = v ( e D M S ) w i t h Aih = r. i h ; 'kh = 'kh

f o r e a c h i , k = 1 , 2 , . . . , q - Aih =-rib; h k h - -T- - 'k h

f o r e a c h i , k = q + l , . . . ,p -

Aih - rib; A k h = kh

f o r i = 1 , 2 , . . . , q ,

k = q + l , , , . , p .

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TRIPATHI AND BAHL

Y X * 1 Let poi = h C i h 3 0 i h / ~ h Cih w i t h C. =(- -I),!{ s2 1 vh h h i

b e t h e w e i q h t e d a v e r a g e o f t h e s t r a t a p o p u l a t i o n re- ; r e s s i c n c o e f f i c i e n t s " - 2

p o i h - Shoi/shi o f y on x . a n d 1

w h e r e P - /S S i s t h e c o r r e l a t i o n c o e f f i - h i k - 'hik h i hk c i e n t b e t w e e n x; and x,, i n s t r a t u r r i h.

I R

F o r p = 1, when i n f o r p l a t i o n on o n l y x i s u s e d , f o l l o w - i i n g I g e and T r i p a t 1 , i (1997) tk ,e optimum v a l u e o f hi i n

( 2 . 7 ) i s y i v e n by

- 'oi - P o i

,Irhen t h e c h o i c e s h. = roi a r ~ 1 t h e r e s u l t i n j v a r i a n c e i s ~ i v c n by

made f o r e a c h i,

w h e r e B = ( b i k ) i , k = 1,. ..,p n

F u r t h e r , ~ v k ~ e n opt imum w e i g h t v e c t o r

i s ~ ~ s e d , we o b t a i n

- 1-f 1 -1 -1 1 2 - - [v(eshc) . n I sz+ ; , ( s ~ B 4 ) LW ( - - i ) s h 0 1 0 1 h "h

a = a 0

I n p r a c t i c e , when e x a c t value o f h .: fj i s n o t 01 o i

a v a i l a b l e , i t may b e e s t i m a t e d t h r o u g h

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E S T I M A T I O N O F MEAN U S I N G D O U B L E S A M P L I N G 2597

Using t h e e s t i m a t e d optimum v a l u e s , we may d e f i n e a - combined m u l t i v a r i a t e e s t i m a t o r f o r Y i n DSS by

For l a r g e samples id(e$c) would & g a i n b e g i v e n by

(3 .1 ) .

One may Yn f a c t o b t a i n s i m u l t a n e o u s optimum

v a l u e s of Ti= a . A. ( i = 1 , 2 , . . . ,p ) a s f o l l o w s . 1 1

* * : Le t s = (Sik) J Q = (Q1,Q2, . . . ,Qp) '

* 1 * where Sik = C 'N (- - l )Shik; Qi= soi i , k=0 ,1 ,2 , . . . , p

h h V h

Then 1-f 2 1 ( e D ) = 7 So + ;, (S :~-~T 'Q+T~S*T)

- S* -1 which g i v e s T = To - o p t 4 (3.2

1-f 2 1 and v0(eDMC) = 7 so + , s g 2 ( 1 - k2) , c+ -, R be ing t h e m u l t i p l e c o r r e l a t i o n where R =

c o e f f i c i e n t between yds and (d l ,d2 , . . . ,d ), w i t h - - P d . = Xids- X i . 1

The optimum v a l u e of T may b e e s t i m a t e d by

* s*--l T = Q*; S* = ( S F ~ k ), Q*= (Q; ,..., Q;)'

1 where sTk = C w ' (- -1 ) shik h Vh * 1

and Qi = c w t ( - -1) s h h V h h o i

Using t h e s e e s t i m a t e d v a l u e s , w e may d e f i n e a combined

m u l t i p l e r e g r e s s i o n e s t i m a t o r f o r ? a s

whose v a r i a n c e f o r l a r g e samples i s g i v e n by (3.2).

Fo r s e p a r a t e e s t i m a t o r , when optimum hi,, i s used

s e p a r a t e l y f o r e a c h i ,

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25 98 TRIPATHI AND BAHL

- w h e r e Bh - ( b h i k )

b . = l - P 2 2 h l k h o i - 'hok + ' h ikFhoiPhok .

Bi,% F u r t h e r i f optimum w e i g h t v e c t o r aoh= - i s u s e d ,

s 'Bh g

w e o b t a i n

I n p r a c t i c e when t h e opt imum c h o i c e Aoih = * P p i h may n o t b e made, i t may b e e s t i m a t e d t h r o u g h Poih

- - shoi/shi 2 a n d a s e p a r a t e m u l t i v a r i a t e r e g r e s s i o n -

t y p e e s t i m a t o r f o r y may b e d e f i n e d a s

e ! ; L = x w ' [ y h h-i z a i h p * o i h - ( x i h - q h ) - 1

I t may b e n o t e d t h a t ~(e!:;~) may b e a p p r o x i m a t e d , f o r

l a r g e n i i n e a c h s t r a t a , t h r o u g h t h e e x p r e s s i o n i n

(3.3).

F o r o b t a i n i n g s i m u l t a n e o u s optimum v a l u e s o f Tih - - aihAih ( i = 1 , 2 , . . . , p ) l e t

Th=(Tlh9T2h,. 9 . ) ' ; Sh"(Shik), Q & Q ~ ~ P Q ~ ~ , * - pQhp) ' - p h

w h e r e Qhi - Shoi. We may e x p r e s s

1-f 2 1 2 V(eDMS) = 7 So + z l C W (- - l ) ( S h o - 2 ~ ~ 3 ~ + ~ ; S h Th)

h "h

w h i c h g i v e s Thopt = Toh = & h h

w i t h 2 2

'h = ' h o ( l - R h o ( l , 2 , . .. , p ) 1

where R h o ( l , 2 , . . . , p ) i s t h e m u l t i p l e c o r r e l a t i o n co-

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ESTIMATION OF MEAN USING DOUBLE SAMPLING 2599

e f f i c i e n t be tween y and x ' s i n t h e h-th s t r a t u m . The * -1 * e s t i m a t e d v a l u e 05 Toh i s g i v e n by T h= s h Qh :"here * * * S h = b h i k ) , Qh= ( Q l h ' . . . A ) ' , Qih= S h o i ' p h

U s i n g t h e above e s t i m a t e d optimum v a l u e , a s e p a -

r a t e m c l t i p l e r e g r e s s i o n e s t i m a t o r f o r 7 may b e d e -

f i n e d a s

whose v a r i a n c e , f o r l a r g e s a m p l e s , i s g i v e n by ( 3 , 4 ) .

4. RELATIVE PERFORMANCE OF THE PROPOSED ESTIXIATaRS

From ( 3 . 2 ) we o b s e r v e t h a t i f t h e w e i g h t e d p a r -

t i a l r e g r e s s i o n c o e f f i c i e n t s Toi a r e u s e d a s T . = a . h 1 1 i'

t h e v a r i a n c e o f t h e c o r r e s p o n d i n g e s t i m a t o r would b e

a l w a y s s m a l l e r t h a n t h a t o f t h e c u s t o m a r y e s t i m a t o r Gs* I n p r a c t i c e , however , e x a c t optimum T, may n o t b e

known. Let T = aT0 = o ~ * - l ? , t h e n f o r a n y T we f i n d f r o m

( 1 . 2 ) , (2 .7) and (3 .2) a f t e r some a l g e b r a i c s i n p l i f i c a -

t i o n t h a t

We n o t e t h a t eDLC would b e b e t t e r t h a n yds a s f a r a s

0 < a < 2. I n p r a c t i c e good g u e s s e d v a l u e s T: o f To may

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

d a t a o r p i l o t s u r v e y a n d b e u s e d i n eDhlC \which would b e

b e t t e r t h a n yds i f T: = aTo , 0 < a < 2 . S i m i l a r l y f r o m

( 1 . 2 ) , ( 2 . 8 ) a n d ( 3 . 4 ) we f i n d t h a t eDXtS would b e b e t t e r

t h a n yds i f

From ( 1 . 2 ) and ( 2 . 9 ) we f i n d t h a t a s u f f i c i e n t con-

d i t i o n f o r eRMC t o b e b e t t e r t h a n yds i s g i v e n b y

'ho 1 -- 'hoi CN % > f o r a l l i = 1 , 2 , . . . , p

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2600 TRIPATHI AND BAHL

- If t h e s t r a t a r a t i o s Rib - R i , t h e n t h e c o n d i t i o n re-

d u c e s t o -

w h i c h i s tb . e u s u a i c o n a i t i f ~ n f o r c i t s ton la ry s e p a r a t e

r a t i o e s t i m a t o r t o b e b e t t e r t h a n mean p e r u n i t . S i m i -

l a r l y f r o m (1.2) and (2.10) it f o l l o w s t h a t emlS would

b e b e t t e r t h a n yds i f (4 .2) h o l d s , It may b e n o t e d t h a t

t h e s e p a r a t e r a t i o , r a t i o - c u m - p r o d u c t and r e g r e s s i o n

t y p e e s t i m a t o r s d i s c u s s e d i n S e c t i o n 3 a r e s u i t a b l e

o n l y f o r l a r g e v a l u e s o f nh i n e a c h s t r a t l ~ m .

The mu1 t i v a r i a t e d i f f e t e r l ~ e (Ra j (1965 j j , r n u l t i -

v a r i a t e r a t i o (Khan a n d T r i p a t h i ( 1 9 6 7 ) ) a n d m u l t i v a -

r i a t e - r a t i o - cu in -p roduc t ( ~ a o and Wudholkar (1967) ) - e s t i m a t o r s f o r t h ~ p o p u l a t i o n mean Y i n USnS a r e de-

f i n e d by

- P - - - ybM = C a i a i w h e r e a = y-A. ( x . -x! ) i = i , 2 , . , . ,p i 1 1 1 i=l -

P -

- ( 5 . 1 ) - Y - y h 4 - C aia i w h e r e a = - x ! i - 1 i = 1 , 2 , . , . , p

i=l x i

P - - Y -

yApM= E a ia i w h e r e ai= = X I

i=l i

P a n d E a . = 1. F u r t h e r

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ESTIMATION OF MEAN USING DOUBLE SAMPLING 2601

M(?&) = v(7bM) w i t h hi = Ri = i , k = 1 , 2 , . . , , p ( 5 . 3 )

'i

hl(?ipM) = V(7bM) w i t h hi=Ri: hk=Rk i , k = 1 , 2 ,..., q (5.4)

h. =-R. ; h =-R i , k=q+l , . . . , p 1 k k

A . =Ri ; h =-Rk i = l , 2 , . . . ,q 1 k

k = q + l , . , . , p

where n i s t h e s i z e o f t h e second phase s ample s e l e c -

t e d randomly. I t may b e n o t e d - t h a t e x ~ r e s s i o n i n (5 .2 )

i s v a l i d f o r a l l s ample s i z e s w h i l e t h e e x p r e s s i o n s i n

(5.3) and (5.4) a r e app rox ima te and v a l i d f o r l a r g e

s amples ,

For compar ison , we assurne i n c a s e o f DSS es t ima-

t o r s t h a t sample a l l o c a t i o n t o t h e s t r a t a i s propor-

t i o n a l ( n h a nA, h = 1 , 2 , . . . , L ) t h a t i s

'We o b t a i n t h a t

1 hi(7ipM)- M(eRpMC)= ( - ) ,Ywha'~L3)a

h

where Dkrn)= ( d $ i ) m = 1 , 2 , 3 .

= [ ( h - ) - i ( i h - F i ) ] [(Yh-P)-hh(qh-7$)];

i , k=1 ,2 , . . . , p

( 2 ) = [ Y ~ - R X ~ ~ ] [ y h - ~ k % h ] ; d h i k i , k = 1 , 2 , . . . , p

di:i = [yh-~iz ih] [ y h - ~ k y k h ] f o r i , k=1 ,2 , , . + ,q

= [ITh-~iFih] [ Y h + ~ k R k h ] f o r i = 1 , 2 , , q

k=q+l , . . . ,p

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2602 TRIPATHI AND BAHL

= [?h+~ix ih ] [ v h + ~ k % h ] f o r i , k = q + l , . . . , p

It i s n o t e d t h a t DL1), ,DL2), 0L3) a r e a l l p o s i -

t i v e d e f i n i t e m a t r i c e s . Thus u n d e r p r o p o r t i o n a l a l l o -

c a t i o n ~ f t h e s e c o n d s a m p l e , t h e m u l t i v a r i a t e combined

d i f f e r e n c e , r a t i o a n d r a t i o - c u m - p r o d u c t e s t i m a t o r s i n

DSS a r e a l w a y s b e t t e r t h a n t h e c o r r e s p o ~ d i n g e s t i m a -

t o r s i n USDS.

ACKNOWLEDGEMENTS

T h e a u t h o r s a r e t h a n k f u l t o t h e r e f e r e e f o r

v a l u a b l e s u g g e s t i o n s l e a d i n g t o a b e t t e r p r e s e n t a -

t i o n sf t h e p a p e r . F u r t h e r , t h e s e c o n d a u t h o r ex-

p r e s s e s h e r g r a t i t u d e t o t h e a u t h o r i t i e s o f F, C,

C o l l e g e , H i s a r , H a r y a n a , f o r g r a n t i n g s t u d y l e a v e

and t o P r o f . R,K. T u t e j a f o r p r o v i d i n g f a c i l i t i e s

t o work a t D e p t t , o f M a t h s . , M O D , U n i v e r s i t y .

BIBLIOGRAPHY

C o c h r a n , W.G. ( 1 9 7 7 ) . S a m p l i n g T e c h n i q u e s i 3 i - d Edi- t i o n , New York, Wiley.

I g e , A b e l b. and T r l p a t h l , TOP. j l 9 8 7 j . On d o u b l e s a m p l i n g f o r s t r a t i f i c a t i o n a n d u s e o f a u x i l i a r i n f o r m a t i o n ; .J. I n d . Soc. Agr. S t a t . , 39,191-201.

Khan, S. a n d T r i p a t h i , TOP. ( 1 9 6 7 ) " The u s e o f m u l t i v a r i a t e a u x i l i a r y i n f o r m a t i o n i n d o u b l e s a m p l i n g ; J. I n d . Assoc . , 5, 42-48.

Raj, D, ( 1 9 6 5 ( a ) ) . On method o f u s i n g m u l t i - a u x i l i a r y i n f o r m a t i o n i n s a m p l e s u r v e y s ; J. Anier. S t a t i s t . AsSOC., 60, 270-277.

Rao, J .N .K , ( 1 9 7 3 ) . On d o u b l e s a m p l i n g f o r s t r a t i - f i c a t i o n a n d a n a l y t i c a l s o r v e y s ; B i o r n e t r i k a , 60, 125-133.

Rao, P,S.R,S, and M a d h o l k a r , G.S, ( 1 9 6 7 ) . Genera - l i z e d m u l t i v a r i a t e e s t i m a t o r s f o r t h e mean of a f i n i t e p o p u l a t i o n ; J . Amer. S t a t i s t . Assoc . , 62, 1009-1012,

Received December 1990; Revised A p r i l 1991.

Recommended Anonymously.

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