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
( 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. (978 79 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.
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
( ( ( ( ( ( ( ( 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
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
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 (..
. 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
( ( 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
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. ( 99-9.. 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 6.9..76..866.89..9.6.79.9.8.8..7.77.6.8..667.96.989 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.
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. 7-6. Krgyra B. (98: A cha- rao yp maor f populao doubl amplg ug wo aulary varabl. Mrka 7 7-. 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. ( 69-76. gh R.K. (98: O mag rao ad produc of populao paramr. al. a. Aoc. Bull. 7-6. 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. 7-6. 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 76 88-8. 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-.