OCTOGON MATHEMATICAL MAGAZINE Vol. 7, No., October 009, pp 667-67 ISSN -5657, ISBN 97-973-55-5-0, www.hetfalu.ro/octogo 667 Improved expoetial estimator for populatio variace usig two auxiliar variables Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache 6 ABSTRACT. I this paper expoetial ratio ad expoetial product tpe estimators usig two auxiliar variables are proposed for estimatig ukow popoulatio variace S. Problem is exteded to the case of two-phase samplig. Theoretical results are supported b a empirical stud. INTRODUCTION It is commo practice to use the auxiliar variable for improvig the precisio of the estimate of a parameter. Out of ma ratio ad product methods are good examples i this cotext. Whe the correlatio betwee the stud ad the auxiliar variate is positive high ratio method of estimatio is quite effective. O the other had, whe this correlatio is egative high product method of estimatio ca be emploed effectivel. Let ad x, deotes the stud variate ad auxiliar variates takig the values i ad x i, z i respectivel, o the uit U i =,,..., N, where x is positivel correlated with ad z is egativel correlated with. To estimate it is assumed that S = N N i, S x = N N xi X ad S z = N N zi Z 6 Received: 7.03.009 000 Mathematics Subject Classificatio. 03E05 Ke words ad phrases. Auxilar iformatio, expoetial estimator, mea squared error
66 Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache are kow. Assume that populatio size N is large so that the fiite populatio correctio terms are igored. Assume that a simple radom samle of size is draw without replacemet SRSWOR from U. The usual ubiased estimator of S is where = s = i. i is the sample mea of. Whe the populatio variace S x = N 93 proposed a ratio estimator for S as xi X is kow, Isaki where s x = t k = s Sx s x xi X is a ubiased estimator of S x.. Upto the first order of approximatio, the variace of S ad MSE of t k igorig the fiite populatio correctio fpc term are respectivel give b where δ pqr = µ p/ µ pqr var s S = MSE t k = 00 µq/ 00 µr/ 00 S, µ pqr = N θ 00 ].3 θ 00 + θ 00 θ 0 ]. i Y p xi X q zi Z r ; p, q, r beig the o-egative itegers. Followig Bahl ad Tuteja 99, we propose expoetial ratio tpe ad expoetial product tpe estimators for estimatig populatio variace S as S t = s exp x s ] x Sx + s x.5
Improved expoetial estimator for populatio variace usig two auxiliar variables 669 s t = s exp z Sz ] s z + Sz.6. BIAS AND MSE OF PROPOSED ESTIMATORS To obtai the bias ad MSE of t, we write s = S + e 0, s x = Sx + e Such that E e 0 = E e = 0 ad E e 0 = θ 00, E e = θ 00, E e 0 e = θ 0. After simplificatio we get the bias ad MSE of t as B t = S MSE t = S To obtai the bias ad MSE of t, we write θ00 θ 0 + 3 ] θ 00 + θ 00 θ 0 + ].. Such that E e 0 = E e = 0 s = S + e 0, s z = S Z + e E e = θ 00, E e 0 e = θ 0 After simplificatio we get the bias ad MSE of t as B t = S MSE t = S θ00 θ 0 5 ] θ 00 + θ 00 + θ 0 9 ].3.
670 Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache 3. IMPROVED ESTIMATOR Followig Kadilar ad Cigi 006 ad Sigh et. al 007, we propose a improved estimator for estimatig populatio variace S as - t = s α exp { S x s x S x + s x } { s + α exp z Sz }] s z + Sz where α is a real costat to be determied such that the MSE of t is miimum. Expressig t i terms of e s, we have { α exp e + e t = S + e 0 } { e + α exp + e }] Expadig the right had side of 3. ad retaiig terms upto secod power of e s, we have t = S + e 0 + e + e + e 0e + α e + e α e + e + +e 0 α e ] + e e αe 0 + e Takig expectatios of both sides of 3.30 ad the substractig S from both sides, we get the bias of the estimator t, upto the first order of approximatio, as B t = S From 3. we have α θ α α 00 + θ 00 + α ] θ 0 t S = S e 0 αe ] α + e θ 0 3. 3. 3.3 3. 3.5 Squarig both the sides of 3.5 ad the takig expectatio, we get MSE of the estimator t, up to the first order of approximatio, as
Improved expoetial estimator for populatio variace usig two auxiliar variables 67 MSE = S θ 00 + α θ 00 + α θ 0 + α θ 0 α α α θ 00 ] θ 0 Miimizatio of 3,6 with respect to α ields its optimum value as α = {θ 00 + θ 0 + θ 0 + θ 0 6} θ 00 + θ 00 + θ 0 3.6 = α 0 sa 3.7 Substitutio of α 0 from 3.70 ito 3.6 gives miimum value of MSE of t.. PROPOSED ESTIMATORS IN TWO PHASE SAMPLING I certai practical situatios whe S x is ot kow a priori, the techique of twophase or double samplig is used. This scheme requires collectio of iformatio o x ad z the first phase sample s of size < N ad o for the secod phase sample s of size < from the first phase sample. The estimators t, t ad t i two-phase samplig will take the followig form, respectivel t d = s k exp s t d = s exp x s ] x s x + s x ] s t d = s z exp { s x s x s x + s x z s z s z + s z } + k exp To obtai the bias ad MSE of t d, t d, t d, we write { s z s z s z + s z }] s = S + e 0, s x = Sx + e, s x = Sx + e...3 where s z = S z + e, s z = Sz + e s x = xi x, s z = zi z
67 Also, Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache x = x i, z = z i E e = E e = 0, E e = θ 00, E e = θ 00, E e e = θ 0 Expressig t d, t d, ad t d i terms of e s ad followig the procedure explaied i sectio ad sectio 3 we get the MSE of these estimators, respectivel as - MSE t d = S θ 00 + θ 00 + + ] θ 0 MSE t d =. = S θ 00 + θ 00 ] θ 0 MSE t d = S + k θ 00 + k θ 00 + k θ 00 + θ 0 + + k k k θ 0 ] θ 0 Miimizatio of.6 with respect to k ields its optimum value as.5.6 k = {θ 00 + θ 0 + θ 0 6} θ 00 + θ 00 + θ 0 = k 0 sa.7 Substitutio of k 0 from.7 to.6 gives miimum value of MSE of t d.
Improved expoetial estimator for populatio variace usig two auxiliar variables 673 5. EMPIRICAL STUDY To illustrate the performace of various estimators of S, we cosider the data give i Murth 967, p.-6. The variates are: : output, x: umber of workers, z: fixed capital, N = 0, = 5, = 0. θ 00 =.667, θ 00 = 3.65, θ 00 =.66, θ 0 =.3377, θ 0 =.0, θ 00 = 3. The percet relative efficiec PRE of various estimators of S with respect to covetioal estimator s has bee computed ad displaed i table 5.. Table 5.: PRE of s, t, t ad mi. MSE t with respect to s Estimator P RE., s s 00 t.35 t.90 t 5.7 I Table 5. PRE of various estimators of s i two-phase samplig with respect to S are displaed. Table 5.: PRE of s, t d, t d ad mi. MSE t d with respect to s Estimator P RE., s s 00 t d 70.76 t d 53.6 t d 7.77 6. CONCLUSION From table 5. ad 5. we ifer that the proposed estimators t performs better tha covetioal estimator s ad other metioed estimators. REFERENCES ] Isaki, C.T., 93, Variace estimatio usig auxiliar iformatio, Joural of America Statistical Associatio. ] Bahl, S. ad Tuteja, R.K., 99, Ratio ad Product tpe expoetial estimator, Iformatio ad Optimizatio scieces, Vol. XII, I, 59-63.
67 Rajesh Sigh, Pakaj Chauha, Nirmala Sawa ad Floreti Smaradache 3] Kadilar, C. ad Cigi, H.,006, Improvemet i estimatig the populatio mea i simple radom samplig, Applied Mathematics Letters, 9006, pp. 75-79. ] Sigh, R., Chauha, P., Sawa, N., ad Smaradache F., 007, Auxiliar iformatio ad a priori values i costructio of improved estimators, Reaissace high press, USA. 5] Murth, M.N., 967, Samplig theor ad methods, Statistical Publishig Societ, Calcutta, Idia. Departmet of Statistics, Baaras Hidu Uiversit U.P., Idia School of Statistics, DAVV, Idore M.P., Idia Departmet of Mathematics, Uiverisit of New Mexico, Gallup, USA