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On non-negativity of the nearest proportional to size sampling design

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COMMUN. STA TIST.—THEORY M ETH., 25(8), 1757-1768 (1996)

O N N O N - N E G A T I V I T Y OF T H E N E A R E S T P R O P O R T I O N A L T O SIZE S A M P L I N G D E S I G N

A R U N K U M A R A D H I K A R Y I N D I A N S T A T IS T IC A L I N S T I T U T E 2 0 3 , B A R R A C K P O R E T R U N K R O A D

C A L C U T T A - 7 0 0 035, I N D I A

Key Words and Phrases : D u ality th eo rem , Ferkas, L em m a; fin ite p o p u la tio n , nearest proportional to size sam pling design, rejective IP P S sam pling plan.

Abstract

The conditions u n d e r which th e n earest p ro p o rtio n a l to size, sam p lin g design introduced b y G ab le r (1987) tu rn s o u t to be n o n -n eg ativ e are identified and these c o n d itio n s are u tilized in g ettin g a rejective IP P S sam pling plan.

1. In tro d u ction

Consider a fin ite p o p u la tio n U of size N an d let y,(i = 1 , '. . . , N ) b e th e vilues of a v a ria te y u n d e r enquiry. O ur problem is to e s tim a te th e population

N

total K = o n th e basis of a sam ple s of fixed size n draw n from the i=l

Population w ith a p ro b a b ility p 0{s).

Gabler (1987) h a s in tro d u c e d th e n earest p ro p o rtio n a l to size sam pling

^esign p*(s ) defined as

P’ (s ) = ( 5 2 A-')P°(S) ( h l !

t e s

"'here A's(i = 1 , . . . , N ) are all positive and are given by

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tt0 A = tt' 7Ti 7T

^21 w here 7r =

~o

ir° ir°

n Nl N2

(1.2)

'i n

"2 N

A

\ ' ~ .. , \ n ) an d tv*' — (tt-,*, . . . , tt;v-*), ?r i°(x ,* )'s bein g th e first order inclusion p ro b ab ilitie s for th e sam p lin g design po(s)(p' (s)) a n d 7r,j°’s being th e second o rder inclusion p ro b a b ilitie s for th e pair of u n its for the design Po(s).

G abler ( 1 9 8 7 ) has also discussed how to realize p ' ( s ) s ta rtin g fro m an a rb itra ry fixed sam ple size (n ) design p 0(s) and he has called such a design

N

a n ' p s design w hich satisfies = n.

_i=i

T h e conditions u n d er w hich th e system of non-hom ogeneous linear equa­

tio n s (1.2) is consistent a n d a d m its of a non-negative so lu tio n for A are con­

sidered in th is p ap e r. T h ese co n d itio n s are utilized in g e ttin g a rejective IP P S sam pling plan.

2. C o n d it io n s for n o n -n e g a t iv it y o f p ' ( s)

F irs t of all we note th a t th e sy stem of non-hom ogeneous lin e ar equations

Jo A = is co nsistent if a n d only if R an k ( x 0) — R an k ( 7r07r*).

In case n is n o n-singular, th e sy stem possesses a u n iq u e solution

A = (2-1)

As a necessary and sufficient co n d itio n for n o n -n eg a tiv ity of A we make use of th e F ark as’ L em m a w hich s ta te s th a t if

Jo y < 0 =>• ( j* , y) < 0 for any y

th e n 7r0 A = tt* ad m its of a n o n -n eg ativ e solution for A . H ere ( v m, y ) denotes th e inner p ro d u c t of th e vectors i t* an d y .

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Example 1

n ( n - l ) N ( N - l )

n ( n —1) n ( n —1) N { N -1) AT( /V — 1) '

• n ( n - l ) N ( N -1)

(Tj . . . x'N ) w here n — 1 Nn

N - 1 + N - 1pi , i = 1 Now , v 0 y < o n n ( n - 1) „

s' + S ( w 3 T ) g !' ' < 0 V :

N

, n n ( n — 1) , h ( n ' — 1) ^ , .

~ N { N - l ) ^ i + N { N - \ ) ^ Vi K 1 n N — n n ( n — l ) JL^

N 1 N - 1 Vi + N { N - 1 ) ^ ’ < * N — n n — 1 A

-y. + — r X > . < 0 v z n - r yv - 1»=1

n - 1 iV

\ ' ,v

A - a _ , t - A

v r , L + v _ , i j y i < . o

4 t=i iV 1 i=i

= > (7r*,y) < o.

This is tru e for any y. Hence by F ark as’ L em m a tt0 \ = 7r" a d m its of

cs< -

n°n-negative so lu tio n for A viz. A,- = ^ p„ 1 = 1 , . . . ,'N.

Example 2

N

Let n = 2 a n d p, > 0, £ p, = 1. For a = .{i, j } ; 1

w e d e f i n e p 0 ( s ) = P>

Pj L

J 1

"’here is th e norm alizatio n factor. T he;, we hav e for i = 1 , . . . ,7V

L 2 1 Pi (1 - P .) , < = P i + Pi I T

7

^Pj.

1 # 1 - W,

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a n d tt° =

L 2 l p , p y

Now ~0 = L2 1

Pi (1 — Pi) P1P2 ••• P\Pn

P iP n P2PN ••• Pn(1 — P n )

so that

"o y <

0

=> y, pt( 1 - p , ) + p , ^ P i V i < 0 V i

= > y .p . + 7 3 < 0 V i

1 Pi jV>'

Vi Pi + 0 V i

Pi N PtVi2

i=i . = 1 1 p* 1 .=1 1 P'

= » (**, ~ rsj2/) < 0

T h is is tru e for any y . H ence by Farkas’ L e m m a

ir0

\ = it' a d m its of a n o n -n eg a tiv e solution for A viz. A; = — 1, - • • , N .

In case tt0 \ = tt* does n o t possess a n o n -n e g a tiv e s o lu tio n for A we caJ check it by m eans of th e T h eo rem of A lternatives o r t h e Duality Tb w hich s ta te s th a t one o f th e following two assertio n s is t r u e

(>) A = 7T* has a p ositive solution

(ii) x 0 y > 0 has a solution satisfying (jt*, y) < 0.

3. R e je c t iv e I P P S S a m p lin g P l a n

L et S an d So d enote respectively th e set of all p o s s i b le samples set o f a r b itra ry sam ples. We m ay define a sam pling p l a n p 0 ( s ) which aS51^

zero p ro b ab ility of selection to each of th e a r b itra ry s a m p le s belonging10"

ju s t by re stric tin g ou r plan to S — So as follows:

Po{s)

pU) for s e S — S 0

!- J 2 p(s) sc So

0 o th e rw ise

w here p(s) is an IP P S sam pling plan.

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O b v i o u s l y

p o (s )

is n o longer an IP P S design. So we are now looking for th e n e a r e s t p r o p o r ti o n a l to size sam pling design p*(s) in tro d u c e d by G abler ( 1 9 8 7 ) in t h e se n se t h a t p ' ( s ) minimizes th e d irec ted d ista n c e D ( p 0, p ' ) from t h e d e s i g n p o ( s ) t o p ' ( s ) defined as

M s )

= v O f ) _ 1

T ' P o ( s )

subject to the constraints ^ p’ (s) = ^ = 1,. - ■, Ar where 7rt's (as- sumed positive f o r e a c h i) are th e first o rd e r inclusion p ro b ab ilitie s for th e IPPS sampling p l a n p ( s ) . So th e idea is as follows:

W e a r e try in g t o g e t rid of th e a rb itra ry sam ples Sq ju s t by confining o u r s e l v e s t o SS Q a n d introducing a new design po{s). As a consequence P o ( s ) d e v i a t e s f r o m t h e original IP P S design p(s) so far as th e inclusion p r o b a b i l i t i e s a re c o n c e r n e d . So we are now searching for a design p ’(s) which is a s n e a r a s p o s s i b le t o po{s) and at th e sam e tim e achieves th e sam e set of f i r s t o r d e r in c lu sio n p ro b a b ilitie s , 7r<, for th e original IP P S sam p lin g plan.

According to G a b l e r (1987), a solution of th e above m in im izatio n pro b lem

’s given by

P*(s ) =

\ >£5 /

Pr°vided p”(s) is n o n - n e g a tiv e for all s t SSo especially w hen all A'.s are n°n~negative w h i c h h o ld in p ractice w hen all th e u n its of th e p o p u la tio n are evenly d istrib u te d o v e r th e set of a rb itra ry sam ples. T h is is e stab lish ed in Theorem 3 .3 t o follow and is also illu stra te d w ith a num erical ex am ple n last s e c tio n .

~~^Slgni 3.1. If i -So, l ^ en no A = does not possess a n o n -n eg a tiv e prllUi°nf0ri -

First vvc a s s u m e th a t

R an k tt0 = R a n k (tt0 x*)

\ — tt" is co n siste n t.

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T h e n (tt* ,y ) = (jr.y ) = - tt,- - - £ tt;- n n ] + i

= £-n P< " I E nPi

N

w here pt = = ^ 2 x i) anc* x i5(* = 1, * • •, AO are know n size : l

= cpx E w

cpi - (1 - Pi)

(c + \ )pi — 1 < 0 if c < — — 1

pi

C o nsider 7t y =

~ o **

‘ 12

* yvi ;‘N2

"in

•7T0^2N

(i - 1)

( N - i)

C o nsider th e p ro d u ct of th e ith row vector of n 0 a n d y .

c ,r°

^n ~

1 n } * i

C T0

^n ~

(n ~ I K ? n

( c + l) 7 r f —

m r f

= [ » - ( " - l ) W > 0 i f c > ( „ _ 1) (!.j n

C onsider th e p ro d u c t of th e j t h row vector of ttq a n d y ( j ^ i).

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( c + l ) < _ £ _ ( n - l ) i r ?

n n n

< i ± M _ ,» > Oi f c > ^ - L

n Trfj

Thus ir0y > 0 if we choose

n — 1 < mr° — 1 < c 7rO

Again (x',y) < 0 if c < i - 1.

So combining (3.1) an d (3.4) we get

n?r° 1

n — 1 < — — 1 < c < --- 1

~ij P<

Thus 7r0 J/ > 0 h as a solution satisfying (tc",y) < 0. So if i £ second assertion is tru e . Hence by th e T h eo rem of A lte rn a tiv e s ir0 ran not admit of a n o n -negative solution for A •

Sssarkjy.: j ^ e b o u n d s of c are consistent.

J n - 1 < - 1 ==> 7T° < tt'?V j ^ i U n — 1 < j— 1 = > np, < 1 V i

W

n7T° ,

^ - 1 < r - 1p>

= > np,.7T? < 7T°

N N

= » n Pi “ > < E 4

y=i j=i

= > n p , . n < njr°

= > < JT?,

"h'ch is true b ec au se w here x = ^ p (s)

s9i|sE5—5o s€5q

£ as i ^ 5 0 = ^ > TTi as (1 - 1) < 1.

’3'ks

Th£22UbJL2 : If all th e u n its are evenly d istrib u te d over V S - S 0), th e n R a n k 7r0 = /V.

(3.3)

(3.4)

(3.5)

So, th e A = j*

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C o n sid er it =

~o

^0

“ 1 T°2 T °'•in

* 2 1 A

• ^ : = (®i «2 • • • , Ojv), say

^ V 1 n N2 ' ‘ -

W e will show th a t O i , . . . , are linearly independent column vectors so t h a t th e ra n k of th e colum n space of tt0 is N. C onsider a linear combination

of , a N

Ci a i + C2 ce2 + ■ C n a N

Ci [7Tj £1 + 7T°}e 2 + • • • + 7T^j ejv] + ’ • ' + Cjv[7Ti/v£l + 7r2A’£2 + ' ' ' + w here e[ = (1 0 0 • • ■ 0), e'2 = ( 0 1 0 • ■ • 0), ■ • •,

ejv = (0 0 0 • • ■ 1) are N linearly independent vectors

— (Ci7r° + ■ • • + Cj^TViN ) ei -f • • • + ( C i + • • • -f Cj^^%)eN- (3-6) If possible, let a 1, a2, • • • , a N form a linearly d ep e n d en t set of vectors.

T h e n

Ci a 1 + C 2a 2 -f • • • -f- Cn Q ji - 0

=>• A t least one of C i, C 2, • • ■, Cn >s non-zero.

S uppose C i is non-zero and C 2 = ■ ■ ■ — Cn = 0 Now from (3.6)

C i + • ■ ■ + Cn o ^ — 0 => C i ^ i + • • • + Cjvfi’ijv — 0 Cl7T;vi + • • • + Cjv 7T%

as £1, • • •, ejv are lin e arly in d e p en d e n t.

Now from (3.7) an d (3.8) we have

C 1ir1o = 0 , - - - , C 1j r ^ = 0 F rom (3.9) we g e t C ^jt? + • • • + jt^ ) = 0

(3.7)

(3.9)

=> n C i X ° = 0

=> Cl 7T°i = 0 (3.10)

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But if all th e u n its are evenly d istrib u ted over S0(S — S 0), th e n 4- 0, ' i V

So C ,jt? = 0 => C , = 0. (3.11) Thus we arrive a t a contradiction. So n j , -- ■ , q \ are lin earlv indeoendent.

Her.ce rank ( tq ) = N.

Remark 3 . 2 If all th e units are evenly d istrib u te d over S o ( S S o ) then Jo A = x * has a unique solution for A viz A = 7r* w here i r j 1 is th e inverse of x 0.

T heorem 3 .3 If all th e u nits are evenly d istrib u te d over So, th e n tt0 A = tt* admits of a non-negative solution for A .

Proof- Now 7r0 A = 7T* will be consistent if an d only if R ank 7r0 = R ank (ToTr*).

Now by T heorem 3.2, if all th e u n its are evenly d is trib u te d over S o (S —So) th en R ank (7T0 ) = N.

So 7r0 A = tt * adm its of a solution if an d only if R ank (7r0 7t*) = N i.e. R ank ( a !Q2 • • ■, a N it *) = N .

Bu; a j • • •, a N are linearly in dependent. So tt * can be expressed as a linear compound of a i , • • •, a jy.

Let tt*C\ cx i cm ce n

"’here cj, ■ • •, cjv are non-zero scalars.

T hus tt* = J 2 (3 -12)

j=i

Consider th e inequality tt0 y > 0

*■1 TT°2 * 1 N ' ' 3/i

7r“i A A n V2

> 0 7rJVl ir° ~ N 2

. V N

+ X u V i + + 7r?nVn > 0 '

X i i V i + A V 2 + • • + A nVn > 0

A i V i + * N2 V2 + • ■ + x nVn > 0 .

.

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N o w ( j * , y ) = = T l y < ( H cj ^ ] )

i=i 1 = 1 j= i

,V iV

= E cj E y>-l j=i .=i

= E ^ E ^ 0.) i=i .=i

(3.14;

T h u s from (3.13) and (3.14) we find th a t th e inequality n 0 V > 0 does no1 have a so lu tio n satisfying (tt *, y) < 0. Hence by th e T h eo rem o f Alternative!

th e first a ssertio n is tru e i.e. tt0 X = ad m its of a non-negative solutior for A •

R e m a r k 3 .3 If A = (cxc2 ■ ■ ■ c ^ ) ' b e a non-negative so lu tio n for A then it i:

easy to check th a t

7! ' o 2 / < 0 = » ( j * , 2 / ) < 0 f o r any y . T h u s F ark a s’ Lem m a holds here.

4 . A n u m e r i c a l e x a m p l e

S u p p o se th e p o p u la tio n consists of N = 7 villages n u m b e re d 1 to 7. TheK are 35 p ossible sam ples, each of size n = 3 , o u t of w hich th e 14 s a m p l e s con­

s ti tu te th e set So of a rb itra ry samples:

1 2 3 2 4 5

1 2 4 2 5 6

1 3 6 2 6 7

1 3 3 4 5

1 4 6 3 5 7

1 4 7 4 6 7

2 3 5 5 6 7

S uppose th a t th e following p,- values are associated w ith th e seven villages.

0.12, 0.14, 0.15, 0.15, 0.14, 0.17, 0.13.

Since th e p, values satisfy th e condition 1

n N

n — l 1 . .

.

-T < P i < - V

I

1 n

we ap p ly m odified M idzuno - Sen (1952, 1953) schem e to get an IP^

schem e w ith th e revised norm ed size m easures 6i s given b y

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T a b l e 4.1

Rejective IP P S sam pling plan co rresp o n d in g to M odified M idzuno - Sen Schem e

s P*(s) s P*(s )

1 2 5 0.037025 2 3 7 0.0435515

1 2 6 0.0487859 2 4 6 0.0619369

1 2 7 0.0304937 2 4 7 0.0452762

1 3 4 0.037441 2 5 7 0.0451112

1 3 5 0.0381801 3 4 6 0.0589856

1 4 5 0.0397059 3 4 7 0.0449543

1 5 6 0.0515426 3 5 6 0.0630282

1 5 7 0.0321097 3 6 7 0.0560379

1 6 7 0.0447155 4 5 6 0.065549

2 3 4 0.048403 4 5 7 0.0477477

2 3 6 0.0594161

Applying th e m e th o d described in S ection 3, we o b ta in th e rejectiv e IP P S sampling plan p*(s) given in T able 4.1, th a t m atch es th e o rig in al 7T,- values and makes th e prob ab ility of selecting a sam ple belonging to th e a rb itra ry set So of samples exactly equal to zero.

S e m a rk 4.1 In th e above exam ple, all th e u n its are evenly d is tr ib u te d over, 4e arb itrary set So of sam ples w hich ensures a n o n -n eg ativ e solution for A and this enables us to co n s tru c t a n earest p ro p o rtio n al to size sam pling design p' (s) retain in g th e sam e IP P S p ro p erty of th e original design p(s).

B I B L I O G R A P H Y

G abler, S. (1987). T h e n ea re st p ro p o rtio n al to size sam pling design. Co mm.

Statist. - Theor. Met h. , 1 6 N o.4, 1117-1131.

M id zu n o , H . (1952). O n th e sam pling system w ith p ro b ab ility p ro p o r­

tional to sum of sizes. An n . Inst. Statist. M a t h., 3, 99-107.

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S e n , A .R . (1953). On th e e stim a tio n of variance in sam p lin g w ith varyinj p ro b ab ilities. Jour. Ind. Soc. Agri. Statist., 5, 119-127.

References

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