Empirical Study in Finite Correlation Coefficient in Two Phase Estimation

Transcription

1 M. Khoshvisa Griffith Uivrsity Griffith Busiss School Australia F. Kaymarm Massachustts Istitut of Tchology Dpartmt of Mchaical girig USA H. P. Sigh R. Sigh Vikram Uivrsity Dpartmt of Mathmatics ad Statistics Idia F. Smaradach Uivrsity of Nw Mico Dpartmt of Mathmatics Gallup USA. mpirical Study i Fiit orrlatio officit i Two Phas stimatio Publishd i: Florti Smaradach Mohammad Khoshvisa Sukato Bhattacharya (ditors) OMPUTATIONAL MODLING IN APPLID PROBLMS: OLLTD PAPRS ON ONOMTRIS OPRATIONS RSARH GAM THORY AND SIMULATION His Phoi USA 006 ISBN: pp. - 7

2 Abstract This papr proposs a class of stimators for populatio corrlatio cofficit wh iformatio about th populatio ma ad populatio variac of o of th variabls is ot availabl but iformatio about ths paramtrs of aothr variabl (auiliary) is availabl i two phas samplig ad aalys its proprtis. Optimum stimator i th class is idtifid with its variac formula. Th stimators of th class ivolv ukow costats whos optimum valus dpd o ukow populatio paramtrs.followig (Sigh 98) ad (Srivastava ad Jhajj 98) it has b show that wh ths populatio paramtrs ar rplacd by thir cosistt stimats th rsultig class of stimators has th sam asymptotic variac as that of optimum stimator. A mpirical study is carrid out to dmostrat th prformac of th costruc stimators. Kywords: orrlatio cofficit Fiit populatio Auiliary iformatio Variac. 000 MS: 9B8 6P0

3 . Itroductio osidr a fiit populatio U {..i..n}. Lt y ad b th study ad auiliary variabls takig valus y i ad i rspctivly for th ith uit. Th corrlatio cofficit btw y ad is dfid by whr S y X N y S y /(S y S ) (.) N N N ( N ) ( yi Y )( i X ) S ( N ) ( i X ) S y ( N ) ( yi Y ) N i i i N y i i Y N. Basd o a simpl radom sampl of si draw without rplacmt i ( i y i ) i ; th usual stimator of y i is th corrspodig sampl corrlatio cofficit : r s y /(s s y) (.) whr s y ( ) ( yi y)( i ) s ( ) ( i ) s y i y y i i i ( ) ( yi y) Th problm of stimatig i i. i y has b arlir tak up by various authors icludig (Koop 970) (Gupta t. al ) (Wakimoto 97) (Gupta ad Sigh 989) (Raa 989) ad (Sigh t. al. 996) i diffrt situatios. (Srivastava ad Jhajj 986) hav furthr cosidrd th problm of stimatig y i th situatios whr th

4 iformatio o auiliary variabl for all uits i th populatio is availabl. I such situatios thy hav suggs a class of stimators for valus of th populatio ma X ad th populatio variac. y y which utilis th kow S of th auiliary variabl I this papr usig two phas samplig mchaism a class of stimators for i th prsc of th availabl kowldg ( Z ad S ) o scod auiliary variabl is cosidrd wh th populatio ma X ad populatio variac S of th mai auiliary variabl ar ot kow.. Th Suggs lass of stimators I may situatios of practical importac it may happ that o iformatio is availabl o th populatio ma X ad populatio variac S w sk to stimat th populatio corrlatio cofficit y from a sampl s obtaid through a two-phas slctio. Allowig simpl radom samplig without rplacmt schm i ach phas th two- phas samplig schm will b as follows: (i) Th first phas sampl of fid si is draw to obsrv oly i s ( s U ) ordr to furish a good stimats of X ad (ii) Giv obsrv y oly. Lt ( ) i ( ) s i s S. s th scod- phas sampl s ( s ) y ( ) y i i s ( ) ( ). i s i s ( ) ( i ) i i s s of fid si is draw to i s W writ u v s s. Whatvr b th sampl chos lt (uv) assum valus i a boudd closd cov subst R of th two-dimsioal ral spac cotaiig th poit (). Lt h (u v) b a fuctio of u ad v such that h() (.) ad such that it satisfis th followig coditios:

5 . Th fuctio h (uv) is cotiuous ad boudd i R.. Th first ad scod partial drivativs of h(uv) ist ad ar cotiuous ad boudd i R. Now o may cosidr th class of stimators of y dfid by hd r h( u v) (.) which is doubl samplig vrsio of th class of stimators ~ r r f ( u v t ) Suggs by (Srivastava ad Jhajj 986) whr kow. u X v s S Somtims v if th populatio ma X ad populatio variac ad ( S ) X ar S of ar ot kow iformatio o a chaply ascrtaiabl variabl closly rla to but compard to rmotly rla to y is availabl o all uits of th populatio. This typ of situatio has b brifly discussd by amog othrs (had 97) (Kirgyra ). Followig (had 97) o may dfi a chai ratio- typ stimator for y as Z s S d r (.) s s whr th populatio ma Z ad populatio variac kow ad ( ) s ( ) ( ) i s i i s i S of scod auiliary variabl ar ar th sampl ma ad sampl variac of basd o prlimiary larg sampl s si (>). of Th stimator d i (.) may b gralid as d α α α α s s r (.) s Z S 6

6 whr α i ' s (i) ar suitably chos costats. May othr graliatio of d is possibl. W hav thrfor cosidrd a mor gral class of y from which a umbr of stimators ca b gra. Th proposd gralid stimators for populatio corrlatio cofficit y is dfid by r t( u v w a) (.) whr w Z a s S ad t(uvwa) is a fuctio of (uvwa) such that t () (.6) Satisfyig th followig coditios: (i) Whatvr b th sampls (s ad s) chos lt (uvwa) assum valus i a closd cov subst S of th four dimsioal ral spac cotaiig th poit P(). (ii) I S th fuctio t(uvwa) is cotiuous ad boudd. (iii) Th first ad scod ordr partial drivativs of t(uvw a) ist ad ar cotiuous ad boudd i S To fid th bias ad variac of s s y S y S ( + ) X ( + ) ( + ) Z ( + ) s X ( + S w writ ) s ( + S ) s y ( + S y ) ( + s ) such that ( 0 ) ( )( )( )0 ad ( i ) 0 i ad igorig th fiit populatio corrctio trms w writ to th first dgr of approimatio 7

7 8 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } ( ) ( ) ( ) ( ) ( ) { } y y y y y y y y whr ( ) / 00 / 00 / 00 m q p pqm pqm μ μ μ μ ( ) ( ) ( ) ( ) N i m i q i p i pqm Z X Y y N μ (pqm) big o-gativ itgrs. To fid th pctatio ad variac of w pad t(uvwa) about th poit P () i a scod- ordr Taylor s sris prss this valu ad th valu of r i trms of s. padig i powrs of s ad rtaiig trms up to scod powr w hav ( ) ( ) + o y (.7) which shows that th bias of is of th ordr - ad so up to ordr - ma squar rror ad th variac of ar sam. padig ( ) y rtaiig trms up to scod powr i s takig pctatio ad usig th abov pc valus w obtai th variac of to th first dgr of approimatio as

8 Var( ) Var( r) + ( ( y / )[ Bt + Dt y / )[ t + ( + F t t + ( ) t + ) t At t ( t t Bt ) t + At t t ] t t ] (.8) whr t (P) t (P) t (P)ad t (P) rspctivly dot th first partial drivativs of t(uvwa) whit rspct to uvw ad a rspctivly at th poit P () Var(r) / )[ ( / ) (/ )( + + ) {( + ) / }] A { D { 0 0 ( 0 y y + y (.9) ( ( 0 / / y y )} )} B { F { ( ( 0 / / y y )} )} Ay paramtric fuctio t(uvwa) satisfyig (.6) ad th coditios () ad () ca grat a stimator of th class(.). Th variac of t t t t at (.6) is miimid for [ A( ) B ] ( ) ( B A ) ( ) [ D( ) F ] α(say) β (say) γ (say) ( ) 00 ( ) ( ) F D 00 (say) (.0) Thus th rsultig (miimum) variac of is giv by A mi. Var( ) Var( r) ( ) y[ ( y D / ) {( D / ) 00 + ( 00 {( A / ) + ( 00 F} ) B} ] ) (.) 9

9 It is obsrvd from (.) that if optimum valus of th paramtrs giv by (.0) ar usd th variac of th stimator is always lss tha that of r as th last two trms o th right had sids of (.) ar o-gativ. Two simpl fuctios t(uvwa) satisfyig th rquird coditios ar t(uvwa) + α u ) + α ( v ) + α ( w ) + α ( a ) t ( u v w a) u α ( v α w α a α ad for both ths fuctios t (P) α t (P) α t (P) α ad t (P) α. Thus o should us optimum valus of α α α ad α i to b o that th stima to gt th miimum variac. It is attaid th miimum variac oly wh th optimum valus of th costats α i (i) which ar fuctios of ukow populatio paramtrs ar kow. To us such stimators i practic o has to us som gussd valus of populatio paramtrs obtaid ithr through past pric or through a pilot sampl survy. It may b furthr o that v if th valus of th costats usd i th stimator ar ot actly qual to thir optimum valus as giv by (.8) but ar clos ough th rsultig stimator will b bttr tha th covtioal stimator as has b illustra by (Das ad Tripathi 978 Sc.). rducs to If o iformatio o scod auiliary variabl is usd th th stimator hd dfid i (.). Takig i (.8) w gt th variac of hd to th first dgr of approimatio as [ h ( ) + ( ) h () Ah () Bh () h () h () ] Var hd ) Var( r) + y + (.) ( which is miimid for h () [ A( ( ) B ) ] h () ( B ( A ) ) (.) Thus th miimum variac of hd is giv by 0

10 mi.var( hd )Var(r) -( ) A {( A ) } y [ + B ] (.) ( ) It follows from (.) ad (.) that mi.var( )-mi.var( hd ) ( ) y [ D {( D + ( which is always positiv. Thus th proposd stimator 00 ) F} ) ] (.) is always bttr tha hd.. A Widr lass of stimators I this sctio w cosidr a class of stimators of y widr tha (.) giv by gd g(ruvwa) (.) whr g(ruvwa) is a fuctio of ruv wa ad such that g( ) ad g( ) r ( ) Procdig as i sctio it ca asily b show to th first ordr of approimatio that th miimum variac of gd is sam as that of giv i (.). It is to b o that th diffrc-typ stimator r d r + α (u-) + α (v-) + α (w-) + α (a-) is a particular cas of ot th mmbr of i (.). gd but it is. Optimum Valus ad Thir stimats Th optimum valus t (P) α t (P) β t (P) γ ad t (P) giv at (.0) ivolvs ukow populatio paramtrs. Wh ths optimum valus ar substitu i (.) it o logr rmais a stimator sic it ivolvs ukow (α β γ ) which ar fuctios of ukow populatio paramtrs say pqm (p qm 0) ad y itslf. Hc it is advisabl to rplac thm by thir cosistt stimats from sampl valus. Lt ( α β γ ) b cosistt stimators of t (P)t (P) t (P) ad t (P) rspctivly whr

11 [ A( ) ] B [ t ( P) α B ] A t ( β ) ( ) [ D ( ) ] 00 F 00 [ t γ ] F D 00 t ( ) ( P ) ( ) with A [ + ( / r)] B [ + ( / )] r D [ + ( / r)] F [ + ( / )] 0 0 s s / ( ) p / q / m / μ μ μ μ / pqm pqm 0 0 r (.) μ pqm p q ( ) ( yi y) ( i ) ( i ) i m ( / ) i s ( ) ( i ) ( / ) i i i i r s y /( s ys ) s y ( ) ( yi y) s ) i i ( ( ). i W th rplac (α β γ ) by ( α β γ ) i th optimum rsultig i th stimator say which is dfid by r t ( u v w a α β γ ) (.) whr th fuctio t(u) U ( u v w a α β γ ) is drivd from th th fuctio t(uvwa) giv at (.) by rplacig th ukow costats ivolvd i it by th cosistt stimats of optimum valus. Th coditio (.6) will th imply that t(p) (.) whr P ( α β γ ) W furthr assum that

12 ( ) ( t U ( ) t P) α β u ( t U t P) v U P U P ( ) ( t U ( ) t P) γ w ( t U t P) (.) a U P U P ( ) ( t U ( ) t P) ο α 6 ο ( t U t P) U P β U P t ( U ) ( ) t 7 ο 8 ο γ ( t U t P) U P U P padig t(u) about P ( α β γ ) i Taylor s sris w hav r[ t ( P + ( β β ) t ) + ( u ) t 6 ( P ) + ( P ) + ( v ) t ( P ) + ( w ) t ( P ) + ( a ) t ( P ( ) ( ) γ γ t P + ( ) t ( P ) + scod ordr trms] 7 8 ) + ( α α) t ( P (.) ) Usig (.) i (.) w hav r[ + ( u ) α + ( v ) β + ( w ) γ + ( a ) + scod ordr trms] (.6) prssig (.6) i trm of s squarig ad rtaiig trms of s up to scod dgr w hav ( + Takig pctatio of both sids i (.7) w gt th variac of y ) y[ ( 0 ) + α( ) + β ( ) + γ ] (.7) approimatio as to th first dgr of

13 Var( ) Var( r) ( ) + ( y D / ) y A {( D / ) 00 + ( {( A / ) + ( F} ) B} ) which is sam as (.) w thus hav stablishd th followig rsult. (.8) Rsult.: If optimum valus of costats i (.0) ar rplacd by thir cosistt stimators ad coditios (.) ad (.) hold good th rsultig stimator sam variac to th first dgr of approimatio as that of optimum. has th Rmark.: It may b asily amid that som spcial cass: α β γ (i) r u v w a (ii) { + ( α u ) + ( γ w )} r { ( β v ) ( a )} (iii) r[ + ( α u ) + ( β u ) + ( γ w ) + ( a )] (iv) r[ ( α u ) ( β u ) ( γ w ) ( a )] of satisfy th coditios (.) ad (.) ad attai th variac (.8). Rmark.: Th fficicis of th stimators discussd i this papr ca b compard for fid cost followig th procdur giv i (Sukhatm t. al. 98).. mpirical Study To illustrat th prformac of various stimators of populatio corrlatio cofficit w cosidr th data giv i (Murthy 967 p. 6]. Th variats ar: youtput Numbr of Workrs Fid apital N80 0

14 X 8.87 Y 8.68 Z y y Th prct rlativ fficicis (PRs) of stimator r hav b compu ad compild i Tabl.. y d hd with rspct to covtioal Tabl.: Th PR s of diffrt stimators of stimator r hd (or PR(.r) y ) tha r ad Tabl. clarly shows that th proposd stimator hd. (or ) is mor fficit Rfrcs: [] had L. (97) Som ratio-typ stimators basd o two or mor auiliary variabls Ph.D. Dissrtatio Iowa Stat Uivrsity Ams Iowa. [] Das A.K. ad Tripathi T.P. ( 978) Us of Auiliary Iformatio i stimatig th Fiit populatio Variac SakhyaSr [] Gupta J.P. Sigh R. ad Lal B. (978) O th stimatio of th fiit populatio corrlatio cofficit-i Sakhya vol. 0 pp [] Gupta J.P. Sigh R. ad Lal B. (979) O th stimatio of th fiit populatio corrlatio cofficit-ii Sakhya vol. pp.-9.

15 [] Gupta J.P. ad Sigh R. (989) Usual corrlatio cofficit i PPSWR samplig Joural of Idia Statistical Associatio vol. 7 pp. -6. [6] Kirgyra B. (980) A chai- ratio typ stimators i fiit populatio doubl samplig usig two auiliary variabls Mtrika vol. 7 pp. 7-. [7] Kirgyra B. (98) Rgrssio typ stimators usig two auiliary variabls ad th modl of doubl samplig from fiit populatios Mtrika vol. pp. -6. [8] Koop J.. (970) stimatio of corrlatio for a fiit Uivrs Mtrika vol. pp [9] Murthy M.N. (967) Samplig Thory ad Mthods Statistical Publishig Socity alcutta Idia. [0] Raa R.S. (989) ocis stimator of bias ad variac of th fiit populatio corrlatio cofficit Jour. Id. Soc. Agr. Stat. vol. o. pp [] Sigh R.K. (98) O stimatig ratio ad product of populatio paramtrs al. Stat. Assoc. Bull. Vol. pp [] Sigh S. Magat N.S. ad Gupta J.P. (996) Improvd stimator of fiit populatio corrlatio cofficit Jour. Id. Soc. Agr. Stat. vol. 8 o. pp. -9. [] Srivastava S.K. (967) A stimator usig auiliary iformatio i sampl survys. al. Stat. Assoc. Bull. vol. 6 pp. -. [] Srivastava S.K. ad Jhajj H.S. (98) A lass of stimators of th populatio ma usig multi-auiliary iformatio al. Stat. Assoc. Bull. vol. pp

16 [] Srivastava S.K. ad Jhajj H.S. (986) O th stimatio of fiit populatio corrlatio cofficit Jour. Id. Soc. Agr. Stat. vol. 8 o. pp [6] Srivkatarma T. ad Tracy D.S. (989) Two-phas samplig for slctio with probability proportioal to si i sampl survys Biomtrika vol. 76 pp [7] Sukhatm P.V. Sukhatm B.V. Sukhatm S. ad Asok. ( 98) Samplig Thory of Survys with Applicatios Idia Socity of Agricultural Statistics Nw Dlhi. [8] Wakimoto K.(97) Stratifid radom samplig (III): stimatio of th corrlatio cofficit A. Ist. Statist Math vol. pp