A FAMILY OF ALMOST UNBIASED ESTIMATORS FOR NEGATIVELY CORRELATED VARIABLES USING JACKKNIFE TECHNIQUE

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1 STATISTICA, ao LXIV,. 4, 004 A FAMIL OF ALMOST UNBIASED ESTIMATORS FOR NEGATIVEL CORRELATED VARIABLES USING JACKKNIFE TECHNIQUE L.N. Upadhyaya, H.P. Sh, S. Sh. INTRODUCTION It s a commo practce to use aulary formato at the estmato stae for creas the effcecy of the estmators. Out of may rato, reresso ad product methods of estmato are ood eamples ths cotet. If the correlato betwee the study character y ad the aulary character s (hh) postve, rato method of estmato s eerally used. O the other had, f ths correlato s hh but eatve, product method of estmato ca be employed. Cosder a fte populato of N uts u u u N : {,,..., } () Let ad X be the characterstcs tak value y ad respectvely o u (,,..., N ). We deote the populato mea of X by X, whch s assumed to be kow ad the populato mea of by, whch s to be estmated. For estmat, Srvekataramaa (980) ad Badyopadhyay (980) proposed a dual to product estmator as X T y () NX where deote the mea of X for the o-sampled uts, y ad N are the sample meas of y ad respectvely. Us predctve approach advocated by Basu (97), Srvastava (983) evsaed aother estmator for as [ X ( N ) ] X s y y ( NX ) (3)

2 768 L.N. Upadhyaya, H.P. Sh, S. Sh Let a sample of sze be draw wthout replacemet from the populato ad let t be splt to sub-samples each of sze m, where m s a teer. Let (, y ),,,..., be the ubased estmators of ( X, ) based o th sub-sample of sze m. The Jackkfe estmators of the type () ad (3), respectvely, based o -th sub-sample are ve by s y X (4) ad X T y,,,..., (5) where NX N,,...,. (6) The estmators dscussed by Srvekataramaa (980) ad Srvastava (983) are eerally based estmators. I ths paper, a attempt has bee made to reduce the bases of these estmators employ Jackkfe techque developed by Queoulle (956). May almost ubased product-type estmators are obtaed ad eplct epressos for ther varaces are also derved to the frst deree of appromato. The mea of almost ubased estmators s that there s o bas up to terms of order. For detals o the bas reducto from product type estmators, the reader ca also refer to Trpath ad Sh (99) ad Sh (003).. PROPOSED ESTIMATORS Let y T, y the class of estmators for as 5 s, y 3 T, y 4 s ad y5 y ad defe y (7) such that 5 ad where,,,3,4,5 are sutably chose costats.

3 A famly of almost ubased estmators for eatvely correlated varables etc. 769 I order to study the bas property of Ŷ, we have the follow lemma whch ca be easly proved by the procedure adopted Cochra (963) ad Sukhatme ad Sukhatme (970). Lemma.. Uder SRSWOR scheme, the relatve bases of the estmators,,3,4,5, to the frst deree of appromato, are ve by RB( y ) p N RB( y ) N RB( y3 ) p N RB( y4 ) N RB( y 5 ) 0 ŷ, (8) where ( B y ) RB( y ),,,3,4, 5 ; ( p k) C ; p ; N C y k, C S y C y ; S C ; s the correlato coeffcet betwee ad y, X N y ( N ) S ( y ) ad N ( N ) S ( X ). Us the results of (8) (7), t s easy to state the follow theorem. Theorem.. A estmator the class of estmators Ŷ at (7) would be ubased f ad oly f ( p ) h p 0 (9) where 3 4 ( f ) h ( f ) ad f. N Proof. It follows from 5 ad (9) that

4 770 L.N. Upadhyaya, H.P. Sh, S. Sh [ ( ph) ( h) ( p)] (0) 5 3 Us (9) ad (0) (7), we obta a famly of almost ubased estmators for as ( u ) { ( )( h )} y T h T s h s () Remark.. If we set 0, the estmators for as u reduces to the famly of almost ubased ( u ) { ( h)} y T h T () whle for 0 (), we et aother famly of almost ubased estmators as ( u ) { ( h)} y s h s (3) The estmator ( u ) (980) estmator Ŷ T, whle s based o Srvekataramaa (980) ad Badyopadhyay ( u ) s based o Srvastava (983) estmator Ŷ s. May other almost ubased estmators for ca be derved from () ust by putt the sutable values of ad. u Remark.. For ( h), verso of Ŷ T : u reduce to the usual almost ubased Jackkfe ( u ) ( f ) X ( f ) X ( ) ( ) y y (4) ad for ( h), of Ŷ s as ( u ) yelds to the usual almost ubased Jackkfe verso { ( ) } ( u ) ( f ) { X ( N ) } ( f ) X N y y (5) ( ) NX ( ) NX

5 A famly of almost ubased estmators for eatvely correlated varables etc. 77 reduce to the usual ub- For 0, the estmators ased estmator y. ( u ), ( u ) ad ( u ) 3. VARIANCE EXPRESSIONS From () we have V( ) [{ ( )( h)} V( y ) V( d ) V( d ) where ( u ) { ( )( h)} Cov( y, d) h Cov y d { ( )( )} (, ) Cov( d, d )] (6) d h T T ad d h s s. It s well kow uder SRSWOR scheme that ( f ) V( y) C y (7) Assum that appromato T T ad s s, the to the frst deree of ( f ) V( d ) ( h) [ C y p( p k) C ] (8) ( f ) V( d ) ( h) [ C y ( k) C ] (9) ( f ) Cov( d, d ) ( h) [ C y { p ( p) k} C ] (0) ( f ) Cov( y, d ) ( h) [ C y pkc ] () ad

6 77 L.N. Upadhyaya, H.P. Sh, S. Sh ( f ) Cov( y, d ) ( h) [ C y kc ] () ( u ) to the frst deree of ap- Putt (7)-() (6), we et the varace of promato as ( u ) ( f ) V( ) [ C y ( p )( h) C {( p )( h)} k] (3) whch s mmsed for p ( h) k (4) Substtuto of (4) (3) yelds the mmum varace of ( u ) as ( u ) ( f ) mv( ) C y[ ] (5) Thus we proved the follow theorem: Theorem 3.. Up to terms of order -, ( u ) ( f ) V( ) S y[ ] (6) wth the equalty s holds f ad oly f p ( h) k. ( u ) It s terest to remark that the lower boud of the varace at (6) s the varace of the usual based lear reresso estmator, whch depcts that the estmators belo to the class are asymptotcally o more effcet tha the lear reresso estmator. We also ote from (7) ad (5) that the mmum ( ) varace of u ( f ) s o loer more tha S y, the varace of the usual ubased estmator y, sce the quatty [ ] of (7) s o more reater tha oe. Remark 3.. Sett 0 (4), we et the optmum value of as h k ( ) opt (say) (7)

7 A famly of almost ubased estmators for eatvely correlated varables etc. 773 ( ) for whch the varace of u () s least ad equal to mv ( ) (5). Thus the substtuto of (7) () yelds the asymptotcally optmum almost ubased estmator (AOAUE) ( u ) as ( u ) opt ( ) k X kh X ( h) ( h) k y y y (8) wth the varace ve at (5). Remark 3.. For putt 0 (4), we et the optmum value of as h k ( ) opt (say) (9) for whch the varace of ( u ) s mmum. For the optmum value of opt ( u ), we et the AOAUE ( u ) as X ( N ) kh X ( N ) k y y y (30) ( h) ( h) NX ( u ) k opt ( ) NX wth the varace ve at (5). Remark 3.3. The estmators ( u ) opt, ( u ) opt ad ( u ) opt ca be used practce whe k s kow. The value of k ca be obtaed from some earler survey or plot study or the epertse athered due course of tme, for stace see Reddy (974, 978), Saha ad Saha (985) ad Murthy (967, pp ). 4. SIMULATION STUD I the preset vestato of smulato study, we focused to fd the eact results based o fte populatos. We eerated a par of N depedet radom umbers y ad (say),,,..., N, from a subroute VNORM wth PHI=0.7, seed(y) = ad seed() = follow Bratley, Fo ad Schrae (983). For fed S = 30 ad S X = 5, we eerated trasformed varables, ad 00.0 ( ) y y S y S (3) 90 SX (3)

8 774 L.N. Upadhyaya, H.P. Sh, S. Sh for dfferet values of the correlato coeffcet. From the eerated populato we computed populato meas ad X. I Table 4., we selected all possble samples of sze =5 from the populato of sze N=0 for a ve N 0 value of, whch results = samples. From the k th (k=,,...,5504) sample, we obtaed three estmates ad s k lr y ( X ), wth y (33) s k, for = (34) k, for = (35) 3 We used a estmate of the lower boud of the varace for each of these estmates as f ( ) ( Vk h k s y r ), for h=,,3. (36) where s sy r s s y. The the 95% coverae was obtaed by cout how may tmes the true populato mea falls the closed terval wth lmts ve by t ( df ) V ( ) (37) h k k h k out of all possble 5504 samples. The coverae so obtaed has bee preseted Table 4.. It s terest to ote that f the correlato s eatve ad hh, the proposed estmators are foud to perform much better tha reresso estmator. I Table 4. we creased our populato sze to N=5 ad sample sze was kept same =5. The results based o all possble 5330 samples have bee preseted. It s remarkable here that the results preseted these Tables are eact ad hece ca be reproduced at ay tme. I Table 4., we creased the sample sze by oe ut, that s =6 by keep N=5, whch results substatal chae total umber of samples ve by O the bass of smulato, oe ca coclude that t s worth to use the proposed estmators f the correlato s eatve ad hh. It s to be oted that althouh the coverae by the proposed estmator rema less certa cases for eatve hh correlato, but keep md t s ubased estmator at the equal level of precso of the reresso estmator.

9 A famly of almost ubased estmators for eatvely correlated varables etc. 775 TABLE 4. The 95% coverae by three estmators for dfferet values of N, ad dfferet values of correlato coeffcets N=0 ad =5 N=5 ad =5 N=5 ad =6 Ŷlr Ŷlr Ŷlr I the et secto we cosder a smulato study based o real data as suested by oe of the revewer. 5. SIMULATION STUD BASED ON REAL DATA I ths secto, we cosder the problem of estmato of sleep hours wth the help of kow ae of the persos lv a partcular localty or tow. The sleep hours eerally decreases as the people becomes older. Such a data collected from N=30 persos s lsted Sh ad Maat (996), pa. 87. A summary of the complete data s ve below: TABLE 5. Summary of parameters of the populato of N=30 uts Parameters Ae () Sleep Hours (y) Mea Stadard Error Meda Mode Stadard Devato Sample Varace Kurtoss Skewess Rae Mmum Mamum Sum Cout

10 776 L.N. Upadhyaya, H.P. Sh, S. Sh The correlato betwee ae ad sleep hours ths populato s We selected all possble samples each of sze 6 uts from the populato cosst of N 30 uts whch results total of samples. The 95% coverae based o ths smulato s reported Table 5.. We also repeated the epermet by select all possble samples each of sze =7 uts whch results total samples. TABLE 5. The 95% coverae by three estmators for N=30 ad dfferet values of Ŷlr The results based o real data shows that the proposed estmator may perform better tha the lear reresso as well as the ubased estmator. The emprcal study was carred out FORTRAN-77 us PENTIUM-0. CONCLUSION The preset vestato provdes a valuable messae for the survey statstcas to deal wth a stuato where eatve correlato ests betwee study ad aulary varables. A lot of efforts have bee made to mprove rato estmator whch works for postve correlato, but oly lmted thouht have bee ve for eatvely correlated varables. The eatvely correlated varables have too much role medcal ad socal sceces. There are several medcal or socal scece related varables whch decreases as the people row up. For eample, as the people become old the follow varables have eatve correlato wth the ae: (a) durato of sleep hours; (b) hear tedecy; (c) eye sht (d) umber of hars o the head; (e) umber of love affars; (f) work hours capacty, ad () amout of blood doato etc. Departmet of Appled Mathematcs Ida School of Mes School of Studes Statstcs Vkram Uversty Departmet of Statstcs St. Cloud State Uversty L. N. UPADHAA HOUSILA P. SINGH SARJINDER SINGH

11 A famly of almost ubased estmators for eatvely correlated varables etc. 777 ACKNOWLEDGEMENTS The authors are heartly thakful to the Eecutve Edtor Professor Stefaa Ma ad a referee for ther valuable commets ad ecouraemet to br the oral mauscrpt the preset form. The opo ad results dscussed ths paper are of authors ad ot ecessary of ther sttute(s). REFERENCES S. BANDOPADHA (980), Improved rato ad product estmators, Sakhya, 4, C, D. BASU (97), A essay o the local foudatos of survey sampl, Part I. Foudatos of Statstcal Iferece, V.P. Godambe ad D.A. Sprott edtors, New ork, 97, P. BRATLE, B. L. FOX, L.E. SCHRAGE (983), A Gude to Smulato, Sprer-Verla, New ork. W.G. COCHRAN (963), Sampl Techques, Joh Wley ad Sos: New ork. M.N. MURTH (967), Sampl theory ad methods, Statstcal Publsh Socety, Calcutta. M.H. QUENOUILLE (956), Notes o bas estmato, Bometrka, 43, V.N. REDD (974), A trasformed rato method of estmato, Sakhya, C, 36, V.N. REDD (978), A study o the pror kowlede o certa populato parameters, Sakhya, C, 40, A. SAHAI, A. SAHAI (985), O effcet use of aulary formato, Joural of Statstcal pla ad ferece,, 03-. R. SINGH, N. S. MANGAT (996), Elemets of survey sampl, Kluwer Academc Publshers, The Netherlads. S. SINGH (003), Advaced sampl theory wth applcatos: how Mchael selected Amy. pp. - 47, Kluwer Academc Press, The Netherlads. S.K. SRIVASTAVA (983), Predctve estmato of fte populato mea us product estmator, Metrka, 30, T. SRIVENKATARAMANA (980), A dual to rato estmator sample surveys, Bometrka, 67, P.V. SUKHATME, ad B.V. SUKHATME (970), Sampl theory of surveys wth applcatos, secod edto, Iowa State Uversty Press. T.P. TRIPATHI, H.P SINGH (99), A class of ubased product-type estmators for the mea sutable for postve ad eatve correlato stuatos, Commucatos statstcs - Theory ad methods,, RIASSUNTO Ua classe d stmator quas corrett per varabl aleatore eatvamete correlate basat sul metodo Jackkfe Utlzzado l metodo Jackkfe vee defta ua classe d stmator quas corrett per, la meda d popolazoe della varable d studo. Ne veoo oltre aalzzate le propretà statstche el campoameto casuale semplce seza rpetzoe. Attraverso ua rcerca emprca vee valutata la performace della soluzoe proposta rspetto allo stmatore d reressoe.

12 778 L.N. Upadhyaya, H.P. Sh, S. Sh SUMMAR A famly of almost ubased estmators for eatvely correlated varables us Jackkfe techque Us Jackkfe techque a famly of almost ubased estmators for, the populato mea of the study varable, s suested ad ts propertes aalysed uder smple radom sampl ad wthout replacemet (SRSWOR) scheme. A emprcal vestato has bee doe to show the performace of the proposed ubased stratees over the based reresso estmator.