EMPIRICAL STUDY IN FINITE CORRELATION COEFFICIENT IN TWO PHASE ESTIMATION

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Philip Knight
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1 MPIRIAL TDY I FIIT ORRLATIO OFFIIT I TWO PHA TIMATIO M. Khohva Lcurr Grffh vry chool of Accoug ad Fac Aurala. F. Kaymarm Aa Profor Maachu Iu of Tchology Dparm of Mchacal grg A; currly a harf vry Thra Ira. H. P. gh R gh Profor of ac Vkram vry Dparm of Mahmac ad ac Ida. F. maradach Aoca Profor Dparm of Mahmac vry of w Mco Gallup A. ABTRAT Th papr propo a cla of maor for populao corrlao coffc wh formao abou h populao ma ad populao varac of o of h varabl o avalabl bu formao abou h paramr of aohr varabl (aulary avalabl wo pha amplg ad aaly propr. Opmum maor h cla dfd wh varac formula. Th maor of h cla volv ukow coa who opmum valu dpd o ukow populao paramr.followg gh (98 ad rvaava ad Jhajj (98 ha b how ha wh h populao paramr ar rplacd by hr co ma h rulg cla of maor ha h am aympoc varac a ha of opmum maor. A mprcal udy carrd ou o dmora h prformac of h corucd maor. Kyword: orrlao coffc F populao Aulary formao Varac. M: 9B8 6P. Iroduco odr a f populao {...}. L y ad b h udy ad aulary varabl akg valu y ad rpcvly for h h u. Th corrlao coffc bw y ad dfd by whr /( y (. ( ( y Y ( X ( ( X y ( ( y Y X Y y. Bad o a mpl radom ampl of draw whou rplacm

2 ( y ; h uual maor of h corrpodg ampl corrlao coffc : r /( y (. whr ( ( y y( ( ( y y ( ( y y y. Th problm of mag ha b arlr ak up by varou auhor cludg Koop (97 Gupa. al. ( Wakmoo (97 Gupa ad gh (989 Raa (989 ad gh. al. (996 dffr uao. rvaava ad Jhajj (986 hav furhr codrd h problm of mag h uao whr h formao o aulary varabl for all u h populao avalabl. I uch uao hy hav uggd a cla of maor for whch ul h kow valu of h populao ma X ad h populao varac of h aulary varabl. I h papr ug wo pha amplg mcham a cla of maor for h prc of h avalabl kowldg ( Z ad o cod aulary varabl codrd wh h populao ma X ad populao varac of h ma aulary varabl ar o kow.. Th uggd la of maor I may uao of praccal mporac may happ ha o formao avalabl o h populao ma X ad populao varac w k o ma h populao corrlao coffc from a ampl obad hrough a wo-pha lco. Allowg mpl radom amplg whou rplacm chm ach pha h wo- pha amplg chm wll b a follow: ( Th fr pha ampl ( of fd draw o obrv oly ordr o furh a good ma of X ad. ( Gv h cod- pha ampl ( of fd draw o obrv y oly. L ( y ( y ( ( ( ( (. u W wr v. Whavr b h ampl cho l (uv aum valu a boudd clod cov ub R of h wo-dmoal ral pac coag h po (. L h (u v b a fuco of u ad v uch ha h( (. ad uch ha af h followg codo:. Th fuco h (uv couou ad boudd R.. Th fr ad cod paral drvav of h(uv ad ar couou ad boudd R.

3 ow o may codr h cla of maor of dfd by hd r h( u v (. whch doubl amplg vro of h cla of maor ~ r r f ( u v uggd by rvaava ad Jhajj (986 whr u X v X ar kow. omm v f h populao ma X ad populao varac o kow formao o a chaply acraabl varabl cloly rlad o bu compard o rmoly rlad o y avalabl o all u of h populao. Th yp of uao ha b brfly dcud by amog ohr chad (97 Krgyra (98 8. ad ( of ar Followg had (97 o may df a cha rao- yp maor for a Z d r (. whr h populao ma Z ad populao varac of cod aulary varabl ar kow ad ( ( ( ar h ampl ma ad ampl varac of bad o prlmary larg ampl (>. Th maor d (. may b grald a α α α d r (. Z whr α ' ( ar uably cho coa. May ohr gralao of d pobl. W hav hrfor codrd a mor gral cla of from whch a umbr of maor ca b grad. Th propod grald maor for populao corrlao coffc dfd by d r ( u v w a (. whr w Z a ad (uvwa a fuco of (uvwa uch ha ( (.6 afyg h followg codo: ( Whavr b h ampl ( ad cho l (uvwa aum valu a clod cov ub of h four dmoal ral pac coag h po P(. ( I h fuco (uvwa couou ad boudd. ( Th fr ad cod ordr paral drvav of (uvw a ad ar couou ad boudd To fd h ba ad varac of w wr d α of

4 ( ( ( ( ( ( ( ( y y Z X X uch ha ( ( ( ( ad ( ad gorg h f populao corrco rm w wr o h fr dgr of appromao ( ( ( ( ( ( ( ( ( ( ( ( ( { } ( ( ( ( ( ( ( ( ( ( ( { } ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( { } ( ( ( ( ( { } ( ( ( ( ( { }. whr ( / / / m q p pqm pqm µ µ µ µ ( ( ( ( m q p pqm Z X Y y µ (pqm bg o-gav gr. To fd h pcao ad varac of d w pad (uvwa abou h po P ( a cod- ordr Taylor r pr h valu ad h valu of r rm of. padg powr of ad rag rm up o cod powr w hav ( d ( o (.7 whch how ha h ba of d of h ordr - ad o up o ordr - ma quar rror ad h varac of d ar am. padg ( d rag rm up o cod powr akg pcao ad ug h abov pcd valu w oba h varac of d o h fr dgr of appromao a

5 Var( d Var( r ( ( / [ / [ ( ( A ( B A B D F ( P ] (.8 whr (P (P (Pad (P rpcvly do h fr paral drvav of (uvwa wh rpc o uvw ad a rpcvly a h po P ( Var(r ( / [ ( / (/ ( {( / }] (.9 A { D { ( ( / / } } B { F { ( ( / / } } ] Ay paramrc fuco (uvwa afyg (.6 ad h codo ( ad ( ca gra a maor of h cla(.. Th varac of d a (.6 mmd for [ A( B ] ( P α(ay ( ( B A β (ay ( [ D( F ] ( (ay P γ ( ( ( F D (ay Thu h rulg (mmum varac of d gv by A {( A / B} m. Var( d Var( r ( [ ] ( {( / } D D F ( / ( (. (. I obrvd from (. ha f opmum valu of h paramr gv by (. ar ud h varac of h maor d alway l ha ha of r a h la wo rm o h rgh had d of (. ar o-gav. Two mpl fuco (uvwa afyg h rqurd codo ar (uvwa α u α ( v α ( w α ( a ( α α α α ( u v w a u v w a ad for boh h fuco (P α (P α (P α ad (P hould u opmum valu of α α α ad α d α. Thu o o g h mmum varac. I o b od ha h mad aad h mmum varac oly wh h opmum d

6 valu of h coa α ( whch ar fuco of ukow populao paramr ar kow. To u uch maor pracc o ha o u om gud valu of populao paramr obad hr hrough pa prc or hrough a plo ampl urvy. I may b furhr od ha v f h valu of h coa ud h maor ar o acly qual o hr opmum valu a gv by (.8 bu ar clo ough h rulg maor wll b br ha h covoal maor a ha b llurad by Da ad Trpah (978 c.. If o formao o cod aulary varabl ud h h maor d rduc o hd dfd (.. Takg (.8 w g h varac of hd o h fr dgr of appromao a Var hd Var( r h ( ( h ( Ah ( Bh ( h ( h (. [ ( ] ( whch mmd for h ( [ A( ( B ] h ( ( B ( A (. Thu h mmum varac of hd gv by m.var( hd Var(r -( A {( A [ B} ] (. ( I follow from (. ad (. ha D {( D F} m.var( d -m.var( hd ( [ ] (. ( whch alway pov. Thu h propod maor d alway br ha hd.. A Wdr la of maor I h co w codr a cla of maor of wdr ha (. gv by gd g(ruvwa (. whr g(ruvwa a fuco of ruv wa ad uch ha g( g( ad r ( Procdg a co ca aly b how o h fr ordr of appromao ha h mmum varac of gd am a ha of d gv (.. I o b od ha h dffrc-yp maor r d r α (u- α (v- α (w- α (a- a parcular ca of gd bu o h mmbr of d (..

7 . Opmum Valu ad Thr ma Th opmum valu (P α (P β (P γ ad (P gv a (. volv ukow populao paramr. Wh h opmum valu ar ubud (. o logr rma a maor c volv ukow (α β γ whch ar fuco of ukow populao paramr ay pqm (p qm ad lf. Hc advabl o rplac hm by hr co ma from ampl valu. L ( α β γ b co maor of (P (P (P ad (P rpcvly whr [ A( ] B [ ( P α B ] A ( β ( [ D ( ] F [ γ ] F D ( ( P ( wh A [ ( / r] B [ ( / r] D [ ( / r] F [ ( / r] p / q / m / µ µ µ / ( / pqm pqm µ p q m µ pqm ( ( y y ( ( r /( y y ( ( y y ( / ( ( ( / ( (. (. W h rplac (α β γ by ( α β γ h opmum d rulg h maor d ay whch dfd by d r ( u v w a α β γ (. whr h fuco ( ( u v w a α β γ drvd from h h fuco (uvwa gv a (. by rplacg h ukow coa volvd by h co ma of opmum valu. Th codo (.6 wll h mply ha (P (. whr P ( α β γ W furhr aum ha ( ( α β u v P P

8 ( ( γ w (. a P ( ( ο α 6 ο ( P P β ( ( 7 ο 8 ο γ P P P P padg ( abou P ( α β γ Taylor r w hav d r[ ( β β ( u 6 ( v ( w ( a ( ( γ γ P ( cod ordr rm] 7 8 ( α α (. g (. (. w hav d r[ ( u α ( v β ( w γ ( a cod ordr rm] (.6 prg (.6 rm of quarg ad rag rm of up o cod dgr w hav ( d [ ( α( β ( γ ] (.7 Takg pcao of boh d (.7 w g h varac of d o h fr dgr of appromao a A {( A / B} Var( d Var( r ( ( (.8 {( / } D D F ( / ( whch am a (. w hu hav ablhd h followg rul. Rul.: If opmum valu of coa (. ar rplacd by hr co maor ad codo (. ad (. hold good h rulg maor d ha h am varac o h fr dgr of appromao a ha of opmum. Rmark.: I may b aly amd ha om pcal ca: d

9 α β γ ( d r u v w a ( d { α( u γ( w } r { β( v ( a } ( r[ α( u β( u γ( w ( a ] d (v d r[ α( u β( u γ( w ( a ] of d afy h codo (. ad (. ad aa h varac (.8. Rmark.: Th ffcc of h maor dcud h papr ca b compard for fd co followg h procdur gv ukham. al. (98 ad Gupa. al. ( mprcal udy To llura h prformac of varou maor of populao corrlao coffc w codr h daa gv Murhy [967 P.6]. Th vara ar: youpu umbr of Workr Fd apal 8 X 8.87 Y 8.68 Z y. 9. Th prc rlav ffcc (PR of d hd d wh rpc o covoal maor r hav b compud ad compld Tabl.. Tabl.: Th PR of dffr maor of maor r hd d (or d PR(.r 9.7. y Tabl. clarly how ha h propod maor ha r ad. hd d (or d mor ffc Rfrc: had L. (97: om rao-yp maor bad o wo or mor aulary varabl. publhd Ph.D. drao Iowa a vry Am Iowa. Gupa J.P. gh R. ad Lal B. (978: O h mao of h f populao corrlao coffc-i. akhya 8-9.

10 Gupa J.P. gh R. ad Lal B. (979: O h mao of h f populao corrlao coffc-ii. akhya -9. Gupa J.P. ad gh R. (989: ual corrlao coffc PPWR amplg. Jour. Id. a. Aoc Krgyra B. (98: A cha- rao yp maor f populao doubl amplg ug wo aulary varabl. Mrka Krgyra B. (98: Rgro yp maor ug wo aulary varabl ad h modl of doubl amplg from f populao. Mrka -6. Koop J.. (97: mao of corrlao for a f vr. Mrka - 9. Murhy M.. (967: amplg Thory ad Mhod. acal Publhg ocy alcua Ida. Raa R.. (989: oc maor of ba ad varac of h f populao corrlao coffc. Jour. Id. oc. Agr. a. ( gh R.K. (98: O mag rao ad produc of populao paramr. al. a. Aoc. Bull gh. Maga.. ad Gupa J.P. (996: Improvd maor of f populao corrlao coffc. Jour. Id. oc. Agr. a. 8( -9. rvaava.k. (967: A maor ug aulary formao ampl urvy. al. a. Aoc. Bull. 6-. rvaava.k. ad Jhajj H.. (98: A la of maor of h populao ma ug mul-aulary formao. al. a. Aoc. Bull rvaava.k. ad Jhajj H.. (986: O h mao of f populao corrlao coffc. Jour. Id. oc. Agr. a.8( 8-9. rvkaarma T. ad Tracy D.. (989: Two-pha amplg for lco wh probably proporoal o ampl urvy. Bomrka ukham P.V. ukham B.V. ukham. ad Aok. ( 98: amplg Thory of urvy wh Applcao. Ida ocy of Agrculural ac w Dlh. Wakmoo K.(97: rafd radom amplg (III: mao of h corrlao coffc. A. I. a Mah 9-.

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