MMUN, STATIST.-THKoUY Mi it(. . 2u<7), 2291-229? <
199
x)ON OPTIM ALITY OF THE HORVITZ-THOMPSON ESTIMATOR UNDER MARKOV PROCESS MODEL
A r u n K u m a r Acihikary1 Departssent O f Mathe m a t i c s
U n i v e r s i t y Of Nairobi P.O. B ox 30197 N a i r o b i , Kenya.
K e y W o r d s a n d P h r a s e s : F i n i t e p o pu lation ; Horvitz-Thompwon e s t i m a t o r M ark o v p r o c e s s model ; Varying probability s a m p l i n g .
A B S T R A C T
or a n y v a r y i n g p r o b a b i l i t y s a m p l i n g design th*
r v i t z - T h o m p s o n ( 1 9 5 2 } e s t i m a t o r is shown to be optinal : h i n t h e c l a s s o f all u n b i a s e d estim ators ot a finito
u l a t i o n t o t a l u n d e r a H a r k o v p r o c e s s model.
1. I N T R O D U C T I O N
C o n s i d e r a finit e p o p u l a t i o n U of s i z e N and let y ^ d e n o t e the v a l u e of a v a r i a t e u n d e r s t u d y for the tth unit o b s e r v e d at time t. (t = 1,...,N ). O u r p r o b l e m is to
N
e s t i m a t e the p o p u l a t i o n t o t a l ^ t= ^ y i t on the b a s i s of a 1=1
s a m p l e s dr a w n f rom 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(s).
A s s u m i n g that the v a r i a t e v a l u e s d e p e n d on t i m e t we c o n s i d e r a M a r k o v p r o c e s s m o d e l d e s c r i b e d b e l o w w h i c h is v e r y u s e f u l in m a n y p r a c t i c a l s i t u a t i o n s ,e . g . i n market r e s e a r c h st u d i e s w h e r e the a n a l y s i s of s a l e s f i g u r e s in s u c c e s s i v e we e k s or m o n t h s is an i m p o r t a n t problem.
C o n s i d e r i n g the c l a s s of a ll u n b i a s e d e s t i m a t o r s for Y t>
the H o r v i t z - T h o m p s o n (1 95 2 ) e s t i m a t o r is s h o w n to be o p t i m a l w i t h i n it> in the s e n s e of h a v i n g m i n i m u m e x p e c t e d
d e s i g n v a r i a n c e u n d e r the s a m e m o d e l .
2. T H E R E S U L T S
Let C d e n o t e the cl a s s of all u n b i a s e d e s t i m a t o r s of the f orm
e t = e ( s , y t) = e ( s , y u | ies) (2.1)
s u c h that e t d e p e n d s on o n l y those y i t ’s w h i c h are in the s a m p l e s and
^ e ( s > y t) p(s) = Y t. (2.2)
s
For e a c h u nit i. we a s s u m e the f o l l o w i n g M a r k o v p r o c e s s
model
y u = + z u • (2.3)
where <x. ( la. I < 1) is a c o n s t a n t and {Z , } is a p u r e l y
V 1 v 1 V t
random p r o c e s s w i t h m e a n zero and v a r i a n c e o'2 .We a lso assume t h a t u n d e r the a b o v e m o d e l y \ t and y are i n d e p e n d e n t l y d i s t r i b u t e d for <• * j.
Using t h e b a c k w a r d s h i f t o p e r a t o r B d e f i n e d as
B Jy. = y. for all j, the m o d e l (2.3) can be
U l , t - j
written as
( 1 - a B ) y., = Z , (2.4)
V tt 1.1
so that w e have
1 7
y u ~ l - a B u
I
= f 1 + a B + a zB 2 + . . . ] Z.,
— 2 + a Z + cx Z + . . .
l.t t t , t —1 t V > t - 2
oo
= y a Z. , . (2.5)
Z j i t., t-Jc
k = o ’
which is k n o w n as a m o v i n g a v e r a g e p r o c e s s of i nfini te order .
Writing E (V ) as the o p e r a t o r for e x p e c t a t i o n ( v a rianc e)m m with r e s p e c t to the ab o v e m o d e l , we have
E (y. ) = 0, m \.t i = 1 , . . . ,N (2.6)
and
00
V (y ) = V [ m i.t ml y a k Z t i . t -k. ]
4c = o J
00 V 2k 2
= Z "i °\
k = 0
O'2
= ----
V
* v (2.7)W r i t i n g E (V ) as the o p e r a t o r for e x p e c t a t i o n
p p
(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 p and
w r i t i n g I"Tas the i n c l u s i o n - p r o b a b i l i t i e s ( a s s u m e d positive) for the uni ts we have the f o l l o w i n g the o r e m .
T h e o r e m 2.1
For any s a m p l i n g d e s i g n p and f or a n y e s t i m a t i o r et m the class C,we have
N 2
E V (e,) f-4- -ll. (2.8)
m p 1 i4(l-«*) I n. J
P r o o f . Wr i t i n g the H o r v i t z - T h o m p s o n ( 1 952) e s t i m a t o r as
* * . r
e t = e <s >Xt> = I "7T i.£S i
and f o l l o w i n g G o d a m b e and J o s h i (196 5 ) w e m a y w r i t e any ot h e r e s t i m a t o r e of the c l a s s C as
e t = e (s.yt) = e ( s , y t) + h ( s , y t) = e t + h t w h e r e
0 = E ph ( s , y t) - ^ h ( s , y t) p(s) s
i m p l y i n g
y h(s
,y t) p(s)
= -Jh(s,yt) p(s) (2.9)
ssi s3?<-
T h e n ,
E V (e.) = E V (e*) + E V (h,) + 2E C (e* h,)
p m t. p m t ' p m t p m t t
( C d e n o t i n g c a v a r i a n c e w i t h r e s p e c t to the m o d e l )•
N , y :
= ‘ . [ I T T {
S 3p<s)} ]
b y u s i n g (2.9)= 0, b y i n d e p e n d e n c e of y u and y ^ ’s V i*j.
So, E V ( e ) >p m t E V (e*). p m t (2.10)
F o l l o w i n g G o d a m b e and T h o m p s o n (1977), we can w r i t e E m V o( e .> = E oV m < e .> + m p t p m t p m t - VJ Ym t l>*
where A (e.) = E ( e . ) - E (Y ),
m t m t m t
so that E V (e.) > E V (e*) - V (Y ) (2.11)
m p t p m t m t
b y u s i n g (2.10)
= 5 - \ ( i 4 l = l( l-ot ) l( 1-0^) V J
Next w e c o n s i d e r the f o l l o w i n g t h e o r e m of p r a c t i c a l i m p o r t a n c e .
Theorem 2 .2
F or a n y s a m p l i n g d e s i g n p and for any e s t i m a t o r et in the c l a s s C, we have
E V (e.) 2: E V <e*).m p t m p t '
P r o o f . F r o m the G o d a m b e - T h o m p s o n ( 1 977) f o r m u l a a p p l i e d to et, we w r i t e
E n,V o < e t> = W « ! ) + E oA ^ e I ) - Wm p t p m t p m t m t' and use t he fa ct that A ( e , ) = 0 to obt a i nm t
E m Vm p e .> = E „v m ( e .> - Wt p m t m t
< E V (e ), b y (2.11).m p t
Remark 1 . W e m a y n o t e that u n d e r the p r e s e n t s t a t i o n a r y time s e r i e s m o d e l all the e s t i m a t o r s h ave b e e n s t a n d a r d i z e d to have m o d e l e x p e c t a t i o n zero.
R e m a r k 2 P o s t u l a t i n g v a r i o u s s u p e r - p o p u l a t i o n m o d e l s the s e v e r a l re s u l t s on o p t i m a l i t y of the H o r v i t z - T h o m p s o n (1952) e s t i m a t o r are a v a i l a b l e in the l i t e r a t u r e viz.
G o d a m b e (1955) ,G o d a m b e and J o s h i ( 1 9 6 5 ) and m a n y others w h e r e b o t h the H o r v i t z - T h o m p s o n ( 1 9 5 2 ) e s t i m a t o r a nd its c o m p e t i t o r s are b a s e d on an i n c l u s i o n probabi lity p r o p o r t i o n a l to si ze (IPPS) s a m p l i n g d e s i g n .But it is i n t e r e s t i n g to n o t e that u n d e r the p r e s e n t s t a t i o n a r y time s e r i e s model n e i t h e r the H o r v i t z - T h o m p s o n (1952) estimator nor its c o m p e t i t o r m u s t be b a s e d on an I P P S s a m p l i n g design or even a fixed s a m p l e s ize d e s i g n and we g e t the o p t i m a l i t y of the H o r v i t z - T h o m p s o n ( 1 95 2) e s t i m a t o r for any v a r y i n g p r o b a b i l i t y s a m p l i n g d e s i g n .Also we may n o t e that no r e s t r i c t i o n s on th e m o d e l p a r a m e t e r s are n e e d e d to e s t a b l i s h the o p t i m a l i t y .
A C K N O W L E D G E M E N T
T he a u t h o r is g r a t e f u l to the r e f e r e e for m a n y helpful s u g g e s t i o n s w h i c h s u b s t a n t i a l l y i m p r o v e d on an earlier v e r s i o n of the paper.
B I B L I O G R A P H Y
G o d a m b e ,V .P . ( 1 9 5 5 ) . A u n i f i e d t h e o r y of s a m p l i n g f r o m finite p o p u l a t i o n s .
J .Roy. S t a t i s t . S o c . S e r . B . ■1 7.2 6 9-2 7 8.
G o d a m b e , V . P . a n d J o s h i ,V .M . (1965 ).A d m i s s i b i 1 ity a n d Bayes e s t i m a t i o n in s a m p l i n g fr om f i n i t e p o p u l a t i o n s -I.
A n n . M a t h . S t a t i s t. 3 6 .1707-1722.
G o d a m b e ,V .P .and T h o m p s o n ,M .E . ( 1 9 7 7 ) . R o b u s 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. 4 7 ,129-146.
H o r v i t z ,D .G .and T h o m p s o n ,D .J . ( 1 9 5 2 ) . A g e n e r a l i z a t i o n of 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 f r o m f i n i t e u n iv erses . J . A m e r .S t a t i s t .A s s o c.47,6 63-685 .
Received June 1990; Revised March 1991.
Recommended by P. S. R. S. Rao, University of Rochester, Rochester, NY.
Refereed by James H. Drew, GTE Labs, Waltham, MA.
