Chilea Joural of Statistics Vol. 5, No. 1, April 014, 49 7 Samplig Theory Research Paper Estimatio of the ratio, prouct a mea usig multi auxiliary variables i the presece of o-respose Suil Kumar Alliace Uiversity, Bagalore, Iia Receive: 05 December 01 Accepte i fial form: 8 Jue 013 Abstract This paper aresses the problem of estimatig the populatio ratio, prouct a mea usig multi auxiliary iformatio i presece of o-respose. Some classes of estimators have bee propose with their properties. Asymptotic optimum estimators i the classs have bee ivestigate alog with their mea square error formulae. Further the optimum value epeig upo populatio parameters whe replace from sample values gives the estimators havig the mea square errors of the asymptotic optimum estimators. A empirical stuy is carrie out i the support of the preset stuy. Both theoretical a empirical fiigs are ecouragig a i favour of the preset stuy. Keywors: Stuy variate Auxiliary variate Bias Mea square error No-respose Mathematics Subject Classificatio: Primary 6D05 1. Itrouctio I survey samplig, it is well recogize that the use of auxiliary iformatio results i substatial gai i efficiecy over covetioal estimators, which o ot utilize such iformatio. The problem of estimatio of ratio, prouct a mea usig sigle auxiliary character has bee ealt to great extet by several authors icluig Sigh 1965, Rao 1987, Bisht a Sisoia 1990, Naik a Gupta 1991, Upahyaya a Sigh 1999, Sigh a Tailor 005 a,b a Sigh et al. 007. Further the problem has bee extee by usig supplemetary iformatio o aitioal auxiliary character by various authors such as Cha 1975, Sahoo a Sahoo 1993, Sahoo et al. 1993, Sahoo a Sahoo 1999 a Sigh a Ruiz Espejo 000. Quite ofte iformatio o may supplemetary variables are available i the survey, which ca be utilize to icrease the precisio of the estimate. Olki 1958 has cosiere the use of multi auxiliary variables, positively correlate with the stuy variable to buil up a multi variate ratio estimator of the populatio mea. Sigh 1967 extee Olki s estimator to the case where auxiliary variables are egatively correlate with variate uer stuy. Later various authors icluig Shukla 1965, 1966, Mohaty 1967, Tujeta a Bahl 1991 a Agrawal a Paa 1993, 1994 have use the iformatio o several auxiliary variables i builig up estimators for populatio mea. Khare 1991 has suggeste a geeralize class of estimators for estimatig the ratio of two meas usig multi ISSN: 0718-791 prit/issn: 0718-790 olie c Chilea Statistical Society Sociea Chilea e Estaística http://www.soche.cl/chjs
50 S. Kumar auxiliary characters with kow populatio meas. It is well kow especially i huma surveys that iformatio is geerally ot obtaie from all the sample uits eve after callbacks. The problem of estimatig the parameters such as ratio of two meas, populatio mea a variace whe some observatios are missig ue to raom o respose has bee iscusse by Touteberg a Srivastava 1998, Sigh a Joarer 1998, Sigh S. et al. 000, Sigh a Tracy 001 a Sigh H. P. et al. 003. I case of o-raom o-respose, the problem of estimatio of populatio mea usig iformatio o sigle auxiliary character has bee cosiere by ifferet authors such as El Bary 1956, Sriath 1971, Cochra 1977, Rao 1986, 1987, Khare a Srivastava 1993, 1995, 1997, Tabasum a Kha 004; Tabasum a Kha 006, Khare a Siha 004, 007, Sigh a Kumar 008 a,b, 009 a,b, 010, 011, Kumar et al. 011 a Gamrot 011 have iscusse the problem of estimatig the ratio of two meas usig multi auxiliary characters i the presece of o-respose. I this paper I have suggeste some classes of estimators for ratio, prouct a mea usig multi auxiliary i ifferet situatios a their properties have bee stuie. Coitios for attaiig miimum mea square error of the propose classes of estimators have also bee obtaie. Estimators base o estimate optimum values have bee obtaie with their approximate mea square error. A empirical stuy has bee carrie out i support of the preset stuy.. Notatios a samplig proceure Let y il i = 0, 1 a x jl j = 1,,..., p be the o-egative values of l th uit of the stuy variate y i i = 0, 1 a the auxiliary variates x j j = 1,,..., p for a populatio of size N with populatio meas Y i i = 0, 1 a X j j = 1,,..., p. Whe o-respose occurs, the subsamplig proceure of Hase a Hurwitz 1946 is a alterative to call backs a similar proceures. I this approach, the populatio of size N is assume to be compose of two strata of size N 1 a N = N N 1, of respoets a o-respoets. The iitial simple raom sample of size is raw without replacemet results i 1 respoets a o-respoets. A sub sample of size m = /k, where k > 1 is preetermie, is raw from the o-respoets a through itesive efforts iformatio o the stuy variates y i i = 0, 1 are assume to be obtaie from all of the m uits see, Rao 1983. Thus the estimator for the populatio mea Y i i = 0, 1 of the fiite populatio is y i = 1 /y i1 + /y i, i = 0, 1.1 where y i1 a y i ;i = 0, 1 are the sample meas of the characters y i i = 0, 1 base o 1 a m uits respectively. The estimator y i is ubiase a has variace 1 f V ar y i = Sy i + W k 1 Sy i. where f = /N, W = N /N, Sy i a Sy i are the populatio mea square of the variates y i i = 0, 1 for the etire populatio a for o-respoig group of the populatio. Similarly the estimator x j j = 1,,...p for the populatio mea X j is give by x j = 1 /x j1 + /x j.3
Chilea Joural of Statistics 51 The estimator x j j = 1,,..., p is ubiase a has te variace V ar x j 1 f = Sx j + W k 1 Sx j.4 where Sx j a Sx j = 1,,..., p are the populatio mea square of x j j for the etire populatio a o respoig group of the populatio. Let ˆR α = y 0 /y α 1, y 1 0 eote the covetioal estimator of the populatio parameter R α = Y 0 /Y α 1, Y 1 0, α beig a costat takes values 1,-1,0. For ifferet values of α, the followig hols i for α = 1, ˆR α ˆR 1 = y 0 y = ˆR say is the covetioal estimator of the ratio 1 R α R 1 = Y 0 /Y 1 = R say. ii for α = 1, ˆR α ˆR 1 = y 0 y 1 = ˆP say is the covetioal estimator of the ratio R α R 1 = Y 0 Y 1 = P say. iii for α = 0, ˆR α ˆR 0 = y 0, is the covetioal estimator of the populatio mea Y 0. Let u j = x j, for j = 1,,..., p = x j p, for j = p + 1, p +,..., p a u eotes the X j X j p colum vector of p elemets u 1, u,..., u p. Super fix T over a colum vector eotes the correspoig row vector. Defiig { } y 0 = Y 01 + η 0, y 1 = Y 11 + η 1, ε 0 = ˆR α /Rα 1 η 0 αη 1 α η1 αη 0η 1, ε j = u j 1, j = 1,,..., p a let ε T = ε 1, ε,..., ε p. The to the first egree of approximatio, the followig hols Eε 0 = α [ C y1 αc y1 ρ y0 y 1 C y0 + Wk 1 C y1 αcy1 ρ y0y 1C y0, Eε j = 0 j = 1,,..., p, [ Eε 0 ε j = E [η 0 αη 1 ε j = q αj + Wk 1 q αj, j = 1,,..., p, Eε 0 ε j = q αj, j = p + 1, p +,..., p, [ Eε j ε l = a jl + W k 1 a jl = e jl say, j, l = 1,,..., p, a = a jl = f jl say, j, l = 1,,..., p, p + 1,..., p where a jl = ρ xj x l C xj C xl, a jl = ρ xj x l C xj C xl, Cy i = Sy i /Y i, Cy = i S y /Y i i, i = 0, 1, Cx j = Sx j /X j, Cx = j S x j /X j, j = 1,,..., p, q αj = C xj ρy0x j C y0 αρ y1x j C y1, q αj = C x j ρy0 x j C y0 αρ y1 x j C y1, j = 1,,..., p, ρy0 y 1, ρ yi x j, ρ xj x l, i = 0, 1; j, l = 1,,..., p a ρ y0y 1, ρ yix j, ρ xjx l, i = 0, 1; j, l = 1,,..., p are the correlatio coefficiets betwee y 0, y 1, y i, x j a x j, x l respectively for the etire populatio a for the o-respoig group of the populatio.
5 S. Kumar b T α = a T α, CT α [ 1 f a T α = q α T = q α1, q α,..., q αp 1 f C T α = = a α1, a α,..., a αp, C α1, C α,..., C αp qα T + W k 1 q T α, C αj = q α T, qα T = q α1, q α,..., q αp, [ 1 f, a αj = q αj + W k 1 1 f q αj, j = 1,,..., p, D = q αj, j = 1,,..., p, [ E F F T F which is assume to be positive efiite. The matrices E = e jl p p a F = f jl p p are p p matrices. Now utilizig the multi auxiliary characters with kow populatio meas, I have suggeste a class of estimators for the parameter R α i sectio 3. 3. The class of estimators Suppose o-respose occurs o the stuy variables y 0, y 1, iformatio o the p- auxiliary variables x j, j = 1,,..., p are obtaie from all sample uits i.e. the iitial sample uits, a the populatio meas X j, j = 1,,..., p of p-auxiliary variables are kow. I this situatio we ote that whe suggestig the estimator for the populatio parameter R α, Khare a Siha 007 use oly the iformatio o the sample meas x j, j = 1,,..., p a o the populatio meas X j, j = 1,,..., p of the p-auxiliary variables x j, j = 1,,..., p. However oe ca also obtai the ubiase estimators x j = 1/ x j1 + / x j of the populatio mea X j, j = 1,,..., p without ay extra effort while i the process of obtaiig y i = 1/ y i1 + / y i, i = 0, 1 the ubiase estimators of the populatio meas Y i, i = 0, 1. Thus, i the situatio state above we have two ubiase estimators x j a x j of the populatio mea X j, j = 1,,..., p of the auxiliary variate x j, j = 1,,..., p. With this backgrou author covice to suggest the class of estimators, G α = G ˆR α, u 1, u,..., u p = G ˆR α, u T of the populatio parameter R α. Let e T eote the row vector of p uit elemets. Whatever be the sample chose, let ˆR α, u T assume values i a close covex subset, S of the p + 1 imesioal real space cotaiig the poit R α, e T. Let G ˆR α, u T be a fuctio of ˆR α, u 1, u,..., u p such that G R α, e T = R α for all R α 3.1 a which is cotiuous a boue with cotiuous a boue first a seco orer partial erivatives i S. Defie a class of estimators of the parameter R α as G α = G ˆR α, u 1, u,..., u p = G ˆR α, u T 3. Sice there are oly a fiite umber of samples, the expectatios a mea square error
Chilea Joural of Statistics 53 of the estimator G α exist uer the above coitios. To obtai the mea square error of G α, expa the fuctio G ˆR α, u T i a seco orer Taylor s series G α = G R α, e T + ˆR α R G. α + u e T G 1 R α, e T ˆR α Rα,e T { + 1 ˆR α R G. α ˆR + ˆR α R α u e T G1. α ˆR** α ˆR,u T α } ˆR** + u e T G ˆR** α, u e, u T where ˆR ** α = R α + η ˆR α R α, u = e + η u e, 0 < η < 1; G 1 eotes the p elemets colum vector of first partial erivatives of G. a G eotes p p matrix of seco partial erivatives of G. with respect to u. Substitutig for ˆR α a u i terms of η 0, η 1, ε 0 a ε a usig 3.1, oe ca get { G α =R α + R α 1 + η0 1 + η 1 α 1 } G. ˆR α + ε T G 1 R α, e T + 1 { 1 + η 0 1 + η 1 α 1 } G. ˆR α ˆR** R α,e T α,u T + R α { 1 + η0 1 + η 1 α 1 } ε T G1. ˆR α +ε T G ˆR** α, ε u T ˆR** α,u T α,u T. 3.3 Takig expectatio i 3.3 a otig that the seco partial erivatives are boue, the followig theorem hols. Theorem 3.1 E G α = Rα + o 1 From theorem 3.1, it follows that the bias of the estimator G α is of the orer 1, a hece its cotributio to the mea square error of G α will be of the orer. Now prove the followig result Theorem 3. To the first egree of approximatio, the mea square error of G 1 R α, e T = R α D 1 b α 3.4
54 S. Kumar a the miimum mea square error is give by mi. MSE G α ˆR α R α bt α D 1 b α 3.5 where MSE ˆR α = ˆR α [ + Wk 1 C y0 + α C y 1 αρc y0 C y1 Cy + 0 α Cy αρc 3.6 1 y 0 C y1 is the mea square error of ˆR α to the first egree of approximatio. Proof From 3.3, the MSE G α to the first egree of approximatio is give by MSE G α = E Gα R [ α = E R α η 0 αη 1 G. ˆR α Rα,e T + ε T G 1 R α, e T 3.7 From 3.1 which implies that G. ˆR? = 1. α Rα,e T Thus the expressio 3.7 reuces to MSE G α = E [ R α η 0 αη 1 + ε T G 1 R α, e T, = E [Rα η 0 αη 1 + R α η 0 αη 1 ε T G 1 R α, e T + G 1 R α, e T T εε T G 1 R α, e T, ˆR α + R α b T α G1 R α, e T + G 1 R α, e T T D G 1 R α, e T 3.8 which is miimize for G 1 R α, e T = R α D 1 b α 3.9 Substitutig 3.9 i 3.8, the resultig miimum mea square error of G α mi.mse G α ˆR α R α bt α D 1 b α 3.10 Thus the theorem is prove. Remark 3.1 The class of estimators G α at 3. is very large. If the parameters i the fuctio G ˆRα, u T are so chose that they satisfy 3.4, the resultig estimator will have MSE give by 3.5. A few examples are: i G 1 = ˆR α exp { α T log u }, iii G 3 = ˆR α exp { φ T u e } v G 5 = ˆR { } α / ˆR α φ T u e ii G = ˆR α [ 1 + φ T u e, iv G 4 = ˆR α + φt u e where φ T = φ 1, φ,..., φ p is a vector of p costats. The optimum values of these
Chilea Joural of Statistics 55 costats are obtaie from 3.4. Sice 3.4 ivolves p equatios, take exactly p ukow costats i efiig above estimators of the class. 4. Estimator base o estimate optimum value To obtai the estimator base o estimate optimum, aopt the same proceure as iscusse i Sigh 198 a Srivastava a Jhajj 1983. It is to be metioe that the propose class of estimator G α at 3. will attaie miimum MSE give by 3.5 or 3.10 oly whe the optimum value of the erivatives or costats ivolve i the estimators give by 3.4, which are fuctios of the ukow populatio parameters are use. To use such estimators i practice, oe has to use some guesse values of the parameters i 3.4, either through past experiece or through a pilot sample survey. It may be ote that eve if the values of the costats use i the estimator are ot exactly equal to their optimum values as give by 3.4 or 3.9 but are close eough, the resultig estimator will be better tha usual estimator ˆR α as has bee emostrate by Das a Tripathi 1978. For more iscussio o this poit i coectio with the estimatio of populatio mea the reaer is referre to Srivastava 1966, Murthy 1967, p.35, Rey 1973, 1974 a Srivekataramaa a Tracy 1980. However there are situatios where the exact optimum values of the erivative give by 3.4 or its guesse value may be rarely kow i practice, hece it is avisable to replace it by its estimate from sample values. We suppose that the equatio 3.4 ca be solve uiquely for the p ukow costats i the estimator 3.. The optimum values of these costats will ivolve D 1 b α or may be both D 1 b α a R α, which are ukow. Whe these optimum values are iserte i 3., it o loger remais a estimator sice it ivolves ukow ψ = D 1 b α, a may be also R α. Let ˆψ be a cosistet estimator of ψ compute from the sample ata at ha. The replace ψ by ˆψ a also R α by ˆR α if ecessary, i the optimum G α resultig i the estimator G α say, which will ow be a fuctio of ˆR α, u a ˆψ. Defie G α = G ˆR α, u T, ˆψT 4.1 where the fuctio G ˆR α, u T, ˆψT is erive from the fuctio G ˆRα, u T cite at 3. by replacig the ukow costats i it by the cosistet estimates. The coitio 3.1 will the imply that G ˆR α, e T, b T = R α for all R α, 4. which i turs implies G. ˆR α R α,e T, ψ T = 1 4.3 We further assume that G. u Rα,e T, ψ T = G. u R α,e T, ψ = R T α ψ 4.4
56 S. Kumar a G. ˆψ R α,e T, ψ T = 0 4.5 Expaig the fuctio G R α, u T, ψ T about the poit R α, e T, ψ T i a Taylor s series a usig 4.1 to 4.5, oe get G α = G R α, e T, ψ T + ˆR α R G. α ˆR + u e T G. u α Rα,e T,ψ T T + ˆψ ψ G. ˆψ + seco orer terms, Rα,e T,ψ T Rα,e T,ψ T = R α + R α ε 0 + ε T R α ψ + seco orer terms. 4.6 Sice ˆψ is a cosistet estimator of ψ, the expectatio of the seco orer terms i 4.6 will be o 1 a hece E G α = R α + o 1 From 4.6 oe obtai G α R α = R α η 0 αη 1 R α ε T ψ + seco orer terms. 4.7 Squarig both sies of 4.7 a eglectig terms of ε s havig power greater tha two G α R α = R α η 0 αη 1 + R α ψt εε T ψ R α η 0 αη 1 ε T ψ or [ G α R α = R α η 0 αη 1 + ψ T εε T ψ η 0 αη 1 ε T ψ 4.8 Takig expectatios of both sies i 4.8 oe get the mea square error of G α, to the first egree of approximatio as MSE G α ˆR α Rα bt α D 1 b α which is same as give i 3.5 or 3.10 i.e. MSE G α = mi.mse G α ˆR α Rα bt α D 1 b α 4.9 It may be ote that the followig estimators:
i 1 = ˆR { } α exp ˆψT log u iii 3 = ˆR α { exp ˆψ } T u e, { } v 5 = ˆR α / ˆR α +ψ T u e, etc. Chilea Joural of Statistics 57 ii = ˆR [ α 1 ˆψ T u e, iv 4 = ˆR α ˆψ T u e, are the members of the suggeste class of estimators G α. It ca be show to the first egree of approximatio that the mea square errors of the estimators G j, j = 1 to 5 are same a equals to the MSE G α = mi.mse G α give by 4.9. For ifferet values of α oe ca obtai a class of estimators for ratio, prouct a populatio mea for G α, H α, F α a J α respectively. The results are explaie i the Appeix I, II, III a IV, respectively. Remark 4.1 Populatio meas X 1, X,..., X p are kow, icomplete iformatio o the stuy variates y 0, y 1 a o the auxiliary variates x j j = 1,,..., p. I this case we use iformatio o 1 + m respoig uits o the stuy variates y 0, y 1 a the auxiliary variate x j j = 1,,..., p from the sample of size alog with kow populatio meas X 1, X,..., X p. Thus propose a class of estimators for R α as H α = H ˆR α, ν T 4.10 where ν eotes the colum vector of p elemets ν 1, ν,..., ν p with ν j = x j /X j, j = 1,,..., p; H ˆR α, ν T is a fuctio of ˆR α, ν T such that H R α, e T = R α for all R α 4.11 H. ˆR α R α,e T = 1 4.1 a also satisfies certai coitios similar to those give for the class of estimators G α at 3. a e T eote the row vector of p uit elemets. It ca be show that E H α = Rα + o 1, a to the first egree of approximatio the MSE of H α is give by which is miimize for MSE H α ˆR α + R α a T α H1 R α, e T + H 1 R α, e T T E H 1 R α, e T 4.14 H 1 R α, e T = R α E 1 a α,
58 S. Kumar a thus the resultig miimum mea square error mi.mse H α ˆR α R α at α E 1 a α 4.14 ˆR α Rα [ { +E 1 { q α + Wk 1 q α q α + W k 1 q α where H 1 R α, e T eote the p elemets colum vector of first partial erivatives of H.. } T }, Thus state the followig theorem: Theorem 4.1 Up to terms of orer 1, with equality holig if MSE H α [MSE ˆR α Rα at α E 1 a α, H 1 R α, e T = R α E 1 a α. Remark 4. Populatio meas X 1, X,..., X p are kow, icomplete iformatio o the stuy variates y 0, y 1 a complete iformatio o the auxiliary variates x j j = 1,,..., p. I this case observe that 1 uits respo o the stuy variates y 0, y 1 but there is complete iformatio o the auxiliary variate x j j = 1,,..., p a the populatio meas X 1, X,..., X p are kow. I such a situatio efie a class of estimators for populatio parameter R α as F α = F ˆR α, w T 4.15 where w eotes the colum vector of p elemets w 1, w,..., w p with w j = x j /X j, j = 1,,..., p; F ˆR α, w T is a fuctio of ˆR α, w T such that F R α, e T = R α for all R α, 4.16 F. ˆR α R α,e T = 1 4.17 a also satisfies certai coitios similar to those give for the class of estimators G α at 3. a e T eote the row vector of p uit elemets. It ca be show that
Chilea Joural of Statistics 59 E F α = Rα + o 1, a to the first egree of approximatio, the MSE of F α is give by MSE F α ˆR α + R α C T α F 1 R α, e T + F 1 R α, e T T F F 1 R α, e T 4.18 where F 1 R α, e T is the p elemets colum vector of the partial erivatives of F., C α = C α1, C α,..., C αp, Cαj = q αj, q αj = C xj ρ y0x j C y0 αρ y1 x j C xj, F = f jl p p, f jl = ρ xjx l C xj C xl. The MSE of F α at 4.18 is miimize for F 1 R α, e T = R α F 1 C α 4.19 a thus the resultig miimum mea square error of F α is give by mi.mse F α ˆR α Rα CT α F 1 C α 4.0 1 f ˆR α Rα qα T F 0 1 q α where F 0 = a jl p p a a jl = ρ xj x l C xj C xl. Thus the followig theorem hols. Theorem 4. Up to terms of orer 1, with equality holig if MSE F α [MSE ˆR α Rα CT α F 1 C α, F 1 R α, e T = R α F 1 C α. Remark 4.3 Populatio meas of auxiliary characters are ukow, icomplete iformatio o the stuy variates y 0, y 1 a complete iformatio o the auxiliary variates x j j = 1,,..., p. I this case, I use iformatio o 1 + m respoig uits o the stuy variates y 0, y 1 a complete iformatio o the auxiliary variate x j j = 1,,..., p. Here i formulatio of the estimator, i aitio to x j j = 1,,..., p I also use the iformatio o x j j = 1,,..., p which ca be easily compute while computig y i i = 0, 1. The populatio meas x j j = 1,,..., p of the auxiliary characters x j j = 1,,..., p are ot kow. With this backgrou efie a class of estimators for the parameter R α as J α = J ˆR α, z T 4.1
60 S. Kumar where z eotes the colum vector of p elemets z 1, z,..., z p, super fix T over a colum vector eotes the correspoig row vector, z j = x j /x j, j = 1,,..., p; J ˆR α, z T is a fuctio of ˆR α, z T such that J R α, e T = R α for all R α 4. J. ˆR α R α,e T = 1 4.3 a also satisfies certai coitios similar to those give for the class of estimators G α at 3. a e T eote the row vector of p uit elemets. It ca be show that E J α = Rα + o 1, a to the first egree of approximatio the MSE of J α is give by MSE J α ˆR α R α +R α a α T J 1 R α, e T J 1 R α, e T T M J 1 R α, e T 4.4 where J 1 α Rα, e T eote the p elemets colum vector of the first partial erivatives of J ˆR α, z T with respect to ˆR α about the poit R α, e T ; a T α = a α1, a α,..., a αp, M = m jl p p, a a T α = W k 1 q T α, q αj = C x j m jl = Wk 1 ρ xj x l C xj C xl = Wk 1 a jl, j, l = 1,,..., p. The MSE of J α at 4.4 is miimize for where M 0 = a jl αj = W k 1 q αj ρ y0 x j C y0 αρ y1 x j C y1, j = 1,,..., p,, j = 1,,..., p, J 1 R α, e T = R α M 1 a α = R αm 1 0 q α 4.5 p p, a jl = ρ xj x l C xj C xl. Thus the resultig miimum mea square error of J α is give by mi.mse J α ˆR α R α a α ˆR α Rα W k 1 T M 1 a q α α T M 1 0 q α 4.6 Thus we state the followig theorem. Theorem 4.3 Up to terms of orer 1,
Chilea Joural of Statistics 61 with equality holig if MSE J α [MSE ˆR α Rα a α T M 1 a α J 1 R α, e T = R α M 1 a α. 5. Efficiecy comparisos Note that b T α D 1 b α = a T α E 1 a α + F T E 1 a α C α T A 1 F T E 1 a α C α 5.1 a b T α D 1 b α = Cα T F 1 C α + a α T M 1 a α 5. where A = F F T E 1 F. Thus from 3.10, the result follows mi.mse [ G α ˆR α Rα a T α E 1 a α + F T E 1 T a α C α A 1 F T E 1 5.3 a α C α [ ˆR α Rα Cα T F 1 C α + From 4.3, 4.37, 4.5 a 5.4, oe obtai mi.mse H α mi.mse Gα = a α T M 1 a α 5.4 Rα F T E 1 T a α C α A 1 F T E 1 a α C α 0 5.5 mi.mse T F α mi.mse Gα = R α M 1 a 0 5.6 a α mi.mse J α mi.mse Gα = R α C T α F 1 C α 0 5.7 α Thus from 5.5, 5.6 a 5.7, the followig iequalities hols mi.mse G α mi.mse Hα mi.mse G α mi.mse Fα mi.mse G α mi.mse Jα 5.8 5.9 5.10 From 5.8, 5.9 a 5.10 it follows that the propose class of estimators G α give by 3. is the best i the sese of havig least miimum MSE amog the classes of estimators G α, H α, F α a J α.
6 S. Kumar 6. Empirical stuy To emostrate the performace of the suggeste estimator relative to usual estimator ˆR α with α = 1, cosier a atural populatio ata earlier cosiere by Khare a Siha 007. The escriptio of the populatio is give below: The ata o the physical growth of upper-socio- ecoomic group of 95 school goig chilre of Varaasi uer a ICMR stuy, Departmet of Peiatrics, BHU urig 1983-84 has bee take uer stuy. The first 5% i.e. 4 chilre uits have bee cosiere as o-respose uits. Deote by y 0 : Height i cm of the chilre, y 1 : Weight i kg of the chilre, x 1 : Skull circumferece i cm of the chilre, x : Chest circumferece i cm of the chilre. The require values of the parameters are: Y 0 = 115.956, Y 1 = 19.4968, X 1 = 51.176, X = 55.8611, C y0 = 0.0515, C y1 = 0.15613, C x1 = 0.03006, C x = 0.05860, C y0 = 0.044, C y1 = 0.11, C x1 = 0.0478, C x = 0.054, ρ y0 x 1 = 0.374 ρ y0 x = 0.60, ρ y1 x 1 = 0.38, ρ y1 x = 0.846, ρ y0 x 1 = 0.571, ρ y0 x = 0.401, ρ y1 x 1 = 0.477, ρ x1 x 1 = 0.97, ρ x1 x 1 = 0.570, ρ y0 y 1 = 0.713, ρ y0 y 1 = 0.678 To illustrate results, cosier the ifferece type estimator usig two auxiliary variables: t = ˆR α + α 1 u 1 1 + α u 1 + φ 1 u 1 1 + φ u 1 6.1 where α i s a φ i s, i = 1, are suitably chose costats, u i = x i /X i a u i = x i /X i, i = 1,. For the sake of coveiece, the MSE of t to the first egree of approximatio is give by MSE t ˆR α + αj e j + α 1 α e 1 + R α α j a αj + j=1 j=1 [ φ j C x j +φ 1 φ a 1 j=1 + R α j=1 φ j q αj + { α 1 φ 1 Cx 1 +α φ 1 a 1 +α 1 φ a 1 +α φ Cx } 6. where e j = { q αj = C xj q αj = C xj a αj = { } Cx j + W k 1 Cx j, j = 1, ; e 1 = ρ y0x j C y0 αρ y1 x j C y1 ρ y0 x j C y0 αρ y1 x j C y1 q αj + Wk 1 q αj }, j = 1,. { } a 1 + W k 1 a 1 ;, j = 1, ; a 1 = ρ x1x C x1 C x ; a 1 = ρ x 1 x C x1 C x ;, j = 1, ; Expressio 6. ca also be obtaie from 3.8 just by puttig the suitable values of the erivatives. The MSE at 6. is miimize for
Chilea Joural of Statistics 63 α 10 = R α α 0 = R α 1 φ 10 = R α φ 0 = R α 1 1 6.3 6.4 6.5 6.6 where = [q αρ x1 x C x1 q α1 C x [C x 1 C x 1 ρ x 1 x = [qαρx 1 x Cx 1 qα1cx [C x 1 C x 1 ρ x 1 x, 1 = [q α1ρ x1 x C x q α C x1, [C x1 Cx 1 ρ x 1 x, 1 = [qα1ρx 1 x Cx qαcx 1. [C x1 C x 1 ρ x 1 x Puttig 6.-6.6 i 6.1, we get the asymptotic optimum estimator AOE i the class of estimators t as t 0 = ˆR α + R α The MSE of t 0 MSE t 0 [ u 1 u 1 + 1 u u + u 1 1 + 1 u 1 to the first egree of approximatio is give by ˆR α Rα 1 f 6.7 { } a 1 C x1 a C x +a 1 a C x1 C x 1 ρ x1 x 1 ρ x1 x + W k 1 { a1 C x1 a C x +a1 a C x1 C x 1 ρ x 1 x 1 ρx1 x } = mi.mse t 6.8 where C a 1 = ρ y0 C y0 x 1 C x1 ρ y1 C y1 x 1 C x1, a = ρ y0 C y0 x C x ρ y1 y1 x C x, a 11 = ρ y0x 11 C a = ρ y0 C y1 y0x C ρ x y 1x C. x C y0 C y1 C ρ x1 y 1x 1 C, x1 I practice the optimum values of α 1, α, φ 1 a φ give by 6.-6.6 are ot kow. I such a case it is worth avisable to replace them by their cosistet estimators i 6.8 a thus oe get a estimator base o estimate optimum values as ˆt 0 = ˆR α [ 1 + ˆ u 1 u 1 + ˆ 1 u u + ˆ u 1 1 + ˆ 1 u 1 6.9 where ˆ, ˆ 1, ˆ a ˆ 1 are the cosistet estimators of, 1, a 1 base o the available ata uer the give samplig esig. It ca be easily show to the first egree of approximatio that MSE ˆt 0 = mi.mse t = mi.mse t 6.10
64 S. Kumar where MSE t 0 isgive by 6.8. Further cosier the followig ifferece type estimators: t 1 = ˆR α + α 1 u 1 1 + α u 1 6.11 t = ˆR α + φ 1 u 1 1 + φ u 1 6.1 t 3 = ˆR α + λ 1 z 1 1 + λ z 1 6.13 where z 1 = x 1 /x 1, z = x /x, α i s, φ i s a λ i s, i = 1, are suitably chose costats. To the first egree of approximatio, the miimum MSE of t 1, t a t 3 are respectively give by R [ α aα1 mi.mse t 1 ˆR α e e1 e e1 + a α e1 a α1 a α e 1 for optimum values of α 1 a α give by 6.14 α10 = Rα[aαe1 aα1e } e 1 e e 1 α0 = R α[a α1 e 1 a α e 1, 6.15 e 1 e e 1 mi.mse t ˆR α R α [ C x 1 C x a 1 [ qα1 C x + q α C x1 q α1q α a 1, 6.16 for optimum values of φ 1 a φ give by φ 10 = Rα[qαa [ 1 q α1c x C x 1 C x a φ 0 = R α[q α1 a [ 1 1 q α C x 1 1 C x 1 C x a, 6.17 mi.mse t 3 ˆR α W k 1 R α [ C x 1 C x a [ qα1 1 C x + C q α x1 q α1q α a 1, 6.18 for optimum values of λ 1 a λ give by λ 10 = R α[q α a [ 1 q α1 C x 1 C x 1 C x a λ 0 = R α[q α1 a [ 1 q α C x 1 1 C x 1 C x a. 6.19 Estimators base o estimate values of α 10, α 0, φ 10, φ 0 a λ 10, λ 0 are respectively give by
Chilea Joural of Statistics 65 ˆt 0 1 = ˆR α + ˆα 10 u 1 1 + ˆα 0 u 1 6.0 ˆt 0 = ˆR α + ˆφ 10 u 1 1 + ˆφ 0 u 1 6.1 ˆt 0 3 = ˆR α + ˆλ 10 z 1 1 + ˆλ 0 z 1 6. where ˆα 10, ˆα 0, ˆφ 10, ˆφ 0, ˆλ 10 a ˆλ 0 are the cosistet estimates of the optimum value α10, α 0, φ 10, φ 0, λ 10 a λ 0 respectively base o the ata available uer the give samplig esig. To the first egree of approximatio, it ca be show that MSE MSE MSE ˆt 0 1 ˆt 0 ˆt 0 3 = mi.mse t 1 6.3 = mi.mse t 6.4 = mi.mse t 3 6.5 where mi.mse t 1, mi.mse t a mi.mse t 3 are respectively give by 6.14, 6.16 a 6.18. I have compute the percet relative efficiecies PREs of t 0 or ˆt 0, t 0 1 or ˆt 0 1, t 0 or ˆt 0 a t 0 3 or ˆt 0 3 with respect to usual estimator ˆR α with α = 1 where t 0 1, t0 a t0 3 are respectively the optimum estimators i t 1, t a t 3. The fiigs are give i Table 1. Table 1. Percet relative efficiecies of the estimators with respect to ˆR α with α = 1 for fixe a ifferet values of k. 1/k 1/5 1/4 1/3 1/ ˆR 100.00 100.00 100.00 100.00 Estimator t 0 t 0 1 t 0 t 0 3 or ˆt 0 or ˆt 0 1 or ˆt 0 or ˆt 0 3 368. 35.4 33.44 309.50 40.1 48.64 61.16 8.91 147.86 158.73 175.89 07.44 117.68 114.78 111.13 106.39 It is observe from Table 1 that the percet relative efficiecies of t 0 or ˆt 0 a t 0 3 or ˆt 0 3 ecrease while the percet relative efficiecies of t 0 1 or ˆt 0 1 a t 0 or ˆt 0 icrease with respect to ˆR as the sub-samplig fractio icreases. It has also bee perceive that t 0 or ˆt 0 is the best amog ˆR, t 0 1 or ˆt 0 1, t 0 or ˆt 0 a t 0 3 or ˆt 0 3. Thus, the suggeste estimator t 0 or ˆt 0 is to be preferre for its use i practice, whe the ifferece type estimator usig two auxiliary variables is use.
66 S. Kumar 7. Coclusio I the preset problem, some classes of estimators for ratio, prouct a mea are iscusse by usig multi auxiliary i ifferet situatios i the presece of o-respose a their properties have bee stuie. Coitios for attaiig miimum mea square error of the propose classes of estimators have also bee obtaie. Estimators base o estimate optimum values have bee obtaie with their approximate mea square error. Due to the o-availability of the ata, I have trie to show the performace of the suggeste estimator relative to usual estimator ˆR α with α = 1 for two auxiliary variables. The performace of the suggeste estimator is preferable whe the o-respose occurs o the stuy as well as auxiliary variables. Ackowlegemets Author wish to thak the leare referees for their critical a costructive commets regarig improvemet of the paper Puttig α = 1, 1, 0 i 3. we get i a class of estimators for ratio R as Appeix I G 1 = G ˆR, u T I-1 ii a class of estimators for prouct P as G 1 = G ˆP, u T I- iii a class of estimators for populatio mea Y 0 as G 0 = G y 0, u T I-3 The miimum mea square errors of the estimators G 1, G 1 a G 0 are respectively give by mi.mse G 1 ˆR R b T 1 D 1 b 1 mi.mse G 1 ˆP P b T 1 D 1 b 1 mi.mse G 0 = Var y 0 Y 0b T 0 D 1 b 0 I-4 I-5 I-6 where MSE ˆR [ = R C +C ρ y0 y1 y 0 y 1 C y0 C y1 + Wk 1, I-7 Cy 0 +C y ρ 1 y 0y 1C y0c y1
Chilea Joural of Statistics 67 MSE ˆP [ = P C +C +ρ y0 y1 y 0y 1 C y0 C y1 + W k 1, I-8 Cy 0 +C y 1 +ρ y 0 y 1 C y0 C y1 are the mea square errors of ˆR a ˆP to the first egree of approximatio, respectively, a [ 1 f Var y 0 = S 0 + W k 1 S 0 I-9 where b T 1 = a 11, a 1,..., a 1p, C 11, C 1,..., C 1p, b T 1 = a 11, a 1,..., a 1p, C 11, C 1,..., C 1p, b T 0 = a 01, a 0,..., a 0p, C 01, C 0,..., C 0p, [ a 1j = q 1j + W k 1 q 1j, j = 1,,..., p; [ a 1j = q 1j + W k 1 q 1j, j = 1,,..., p; [ a 0j = q 0j + Wk 1 q 0j, j = 1,,..., p; q 1j = C xj ρy0x j C y0 ρ y1x j C y1, q 1j = C x j ρy0 x j C y0 ρ y1 x j C y1, q 1j = C xj ρy0x j C y0 +ρ y1x j C y1, q 1j = C x j ρy0 x j C y0 +ρ y1 x j C y1, q 0j = ρ y0 x j C y0 C xj, q 0j = ρ y 0 x j C y0 C xj, C 1j = C 1j = q 1j, C 0j = q 0j. q 1j, 8. Appeix II Puttig α = 1, 1, 0 i 4.10 we get the class of estimators i for ratio R as H 1 = H ˆR, ν T II-1 ii for prouct P as H 1 = H ˆP, ν T II- iii for populatio mea Y 0 as H 0 = H y 0, ν T II-3 The miimum mea square errors of the estimators H 1, H 1 a H 0 ca be obtaie from 4.13 by puttig α = 1, 1, 0 a are respectively give by
68 S. Kumar mi.mse H 1 ˆR R a T 1 E 1 a 1 mi.mse H 1 ˆP P a T 1 E 1 a 1 mi.mse H 0 = Var y 0 Y 0a T 0 E 1 a 0 II-4 II-5 II-6 where a 1 = a 11, a 1,..., a 1p, a 0 = a 01, a 0,..., a 0p, a 1 = a 11, a 1,..., a 1p, [ a 1j = q 1j + W k 1 q 1 j a 1j = a 0j = [ [, j = 1,,..., p; q 1j + Wk 1 q 1 j, j = 1,,..., p; q 0j + W k 1 q 0 j, j = 1,,..., p; where q 1j, q 1j, q 0j, q 1j, q 1j, q 0j are same as efie earlier. It is to be metioe that the class of estimators t 1 = ˆR h ν T II-7 of the ratio R reporte by Khare a Siha 007 is a member of the propose class of estimator H 1. To the first egree of approximatio, where mi.mse H 1 is give by II-. mi.mse t 1 = mi.mse H 1 II-8 9. Appeix III Puttig α = 1, 1, 0 i 4.15 we get the class of estimators i for ratio R as F 1 = F ˆR, w T III-1 ii for prouct P as F 1 = F ˆP, w T III- iii for populatio mea Y 0 as F 0 = F y 0, w T III-3 The miimum mea square errors of the estimators F 1, F 1 a F 0 are respectively give by
Chilea Joural of Statistics 69 mi.mse F 1 ˆR R C T 1 F 1 C 1 mi.mse F 1 ˆP P C T 1 F 1 C 1 mi.mse F 0 = Var y 0 Y 0C T 0 F 1 C 0 III-4 III-5 III-6 where C 1 = C 11, C 1,..., C 1p, C0 = C 01, C 0,..., C 0p, C 1 = C 11, C 1,..., C 1p C 1j = q 1j, j = 1,,..., p, C 1j = q 1j, j = 1,,..., p, C 0j = q 0j, j = 1,,..., p, q 1j, q 1j a q 0j, j = 1,,..., p are same as efie earlier. Khare a Siha 007 suggeste a class of estimators for ratio R as t = ˆR f w T III-7 where f w T is a fuctio of w T = w 1, w,..., w p such that f e T = 1. The estimator t is ue to Khare a Siha 007 a member of the class F 1 efie at III-7. The miimum MSE of t is give by mi.mse t = mi.mse F 1 ˆR R C T 1 F 1 C 1 III-8 Appeix IV Puttig α = 1, 1, 0 i 4.1 we get the class of estimators i for ratio R as J 1 = J ˆR, z T IV-1 ii for prouct P as J 1 = J ˆP, z T IV- iii for populatio mea Y 0 as J 0 = J ȳ 0, z T IV-3 The miimum mea square errors of the estimators J 1, J 1 a J 0 are respectively give by
70 S. Kumar mi.mse J 1 ˆR R a 1 T M 1 a 1 mi.mse J 1 ˆP P a T 1 M 1 a 1 mi.mse J 0 ȳ 0 Ȳ 0 a 0 T M 1 a 0 IV-4 IV-5 IV-6 where a 1 = a 11, a 1, a 13,..., a 1p a 0 = a 01, a 0, a 03,..., a 0p a 0j = Wk 1 q 0j, q 1j = C x j ρy0x jc y0 +ρ y1x jc y1, a 1 =, a 1j = Wk 1 q 1j, a 11, a 1, a 13,..., a 1p, a 1j = Wk 1 q 1j, q 1j = C x j ρy0x jc y0 ρ y1x jc y1,, q 0j = ρ y 0x jc y0 C xj, j = 1,,..., p. Refereces Agrawal, M. C., Paa, K. B., 1993. A efficiet estimator i post stratificatio. Metro, Vol. L1, 179-188. Agarwal, M. C., Paa, K. B., 1994. O multivariate ratio estimatio. Jour. I. Statist. Assoc., 3, 103-110. Bahl, S., Tuteja, R. K., 1991. Ratio a prouct type expoetial estimator. Iformatio a optimizatio scieces, 11, 159-163. Bisht, K. K. S., Sisoia, B. V. S., 1990. Efficiet estimators of mea of fiite populatios with kow coefficiet of variatio. Jour. I. Soc. Agril. Statist., 4, 1, 131-139. Cha, L.,1975. Some ratio type estimators base o two or more auxiliary variables. Ph. D. thesis submitte to lowa State Uiversity, Ames., Lowa. Cochra, W.G., 1977. Samplig Techiques, 3r e., Joh Wiley a Sos, New York. Das, A. K., Tripathi, T.P., 1978. Use of auxiliary iformatio i estimatig the fiite populatio variace. Sakhya C, 40, 139-148. El- Bary, M. A., 1956. A samplig proceure for maile questioaires. Jour. Amer. Statist. Assoc., 51, 09-7. Hase, M.H., Hurwitz, W.N., 1946. The problem of o-respose i sample surveys. Jour. Amer. Statist. Assoc., 41, 517-59. Khare, B. B., 1991. O geeralize class of estimators for ratio of two populatio meas usig multi auxiliary characters. Alig. Jour. Statist., 11, 81-90. Khare, B. B., Siha, R. R., 004. Estimatio of fiite populatio ratio usig two-phase samplig scheme i the presece of o-respose. Alig. Jour. Statist., 4, 43-56. Khare, B. B., Siha, R. R., 007. Estimatio of the ratio of the two populatio meas usig multi auxiliary characters i the presece of o-respose. Statistical Techiques i Life Testig, Reliability, Samplig Theory a Quality Cotrol, 163-171. Khare, B. B, Srivastava, S., 1993. Estimatio of populatio mea usig auxiliary character i presece of o-respose. Nat. Aca. Sc. Letters, Iia, 163, 111-114. Khare, B. B, Srivastava, S., 1995. Stuy of covetioal a alterative two-phase samplig ratio, prouct a regressio estimators i presece of o-respose. Proc. Nat. Aca. Sci. Iia, 65A, II, 195-03. Khare, B. B, Srivastava, S., 1997. Trasforme ratio type estimators for the populatio
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