General Families of Estimators for Estimating Population Mean in Stratified Random Sampling under Non-Response

Transcription

1 J. tat. Appl. Pro. Lett. 3, No., Joural of tatstcs Applcatos & Probablty Letters A Iteratoal Joural Geeral Famles of Estmators for Estmatg Populato Mea tratfed Radom amplg uder No-Respose Maoj K. haudhary ad aurabh Departmet of tatstcs, Baaras Hdu Uversty, Varaas, Ida Receved: 4 Jul. 05, Revsed: 5 Aug. 05, Accepted: 7 Aug. 05 Publshed ole: Ja. 06 Abstract: The preset artcle cocetrates o estmatg the mea of a stratfed populato the presece of o-respose. I ths artcle, we have suggested separate ad combed-type famles of estmators of populato mea usg the formato of a auxlary varable assumg that the o-respose s observed o both study ad auxlary varables. The propertes of the suggested famles have bee thrashed out. The suggested famles have bee dscussed uder the proportoal, Neyma ad some other allocato schemes proposed by haudhary et al. 6. A emprcal study has also bee carred out the support of theoretcal results. Keywords: tratfed radom samplg, separate-type famly, combed-type famly, populato mea ad o-respose. Itroducto No-respose s a very serous ssue estmatg the populato parameters through a mal survey. Hase ad Hurwtz were the frst who coped up the problem of o-respose whle coductg mal surveys. They developed a techque of sub-samplg of o-respodets to deal wth the problem of o-respose ad ts adjustmets. I pot of fact, they suggested a ubased estmator of populato mea uder o-respose by dvdg the populatoto two dfferet groups, vz. group of respodets ad group of o-respodets. I order to avod the bas due to o-respose, they suggested a techque of selectg a sub-sample from the o-respodets of the sample. A lot of wors have bee doe for estmatg the populato mea stratfed radom samplg wheever the vestgator suffers wth the problem of o-respose. Khare has dscussed the problem of optmum allocato stratfed radom samplg the presece of o-respose. Kha et al. 3 descrbed the method of optmum allocato multvarate stratfed radom samplg uder o-respose. haudhary et al. 4 have proposed a geeral famly of estmators stratfed radom samplg the presece of o-respose by cosderg Khoshevsa et. al. 5, haudhary et al. 6 have proposed some ew allocato schemes based o respose ad o-respose rates stratfed radom samplg. Further, haudhary et al. 7 have suggested a class of factor-type estmators of populato mea stratfed radom samplg uder o-respose. All of the wors metoed above have bee carred out the stuatos whe o-respose s observed o study varable ad auxlary varables s free from o-respose. But the stuatos whch both study ad auxlary varables are suffered from o-respose, t would be evtable to troduce the estmators of populato parameters of study varable. I the lght of above crcumstaces, we have suggested some famles of factor-type estmators of populato mea stratfed radom samplg usg a auxlary varable uder o-respose. The optmum estmators of the proposed famles have bee dscussed. We have compared the proportoal ad Neyma allocatos wth some other allocato schemes based o respose ad/or o-respose rates through the suggested famles of estmators. The theoretcal study has also bee supported wth umercal aalyss. orrespodg author e-mal: rtamaoj5@gmal.com c 06 NP

2 8 M. K. haudhary, aurabh: Estmators for populato mea stratfed... amplg trategy ad Estmato Procedure Let us suppose that a populato cossts of N uts s dvded to strata. Let there be N uts the th stratum,,...,. A radom sample of sze s selected from the etre populato such a way that uts are selected from the th stratum so that =. It s oted that out of uts there are uts who supply the formato ad uts who do ot respod. Usg Hase ad Hurwtz techque of sub-samplg of o-respodets, a sub-sample of h = /L,L o-respodets s selected from the uts by smple radom samplg wthout replacemet RWOR scheme ad the formatos obtaed o all the h uts. Let X 0 ad X be the study ad auxlary varables respectvely wth ther respectve populato meas X 0 ad X. Thus the Hase ad Hurwtz estmators of X 0 adx are respectvely gve by T0st = Tst = p T 0 p T where T0 = x 0 x 0h, T = x x h, p = N N, x 0 ad x are the meas based o respodet uts for the study ad auxlary varables respectvely. x 0h ad x h are the meas based o h o-respodet uts for the study ad auxlary varables respectvely. The varaces of the ubased estmators T0st ad T st are respectvely gve by VT0st= p L N 0 W p 0 3 ad VT st = N p L W p 4 where 0 ad 0 are the populato mea squares of the etre group ad o-respose group respectvely for study varable th stratum. mlarly ad are the populato mea squares of the etre group ad o-respose group respectvely for auxlary varable th stratum. W s the o-respose rate the th stratum. 3 Proposed Famles of Estmators It s very dffcult to estmate the populato parameters usg auxlary varables uder the stuatos whch both study ad auxlary varables are suffered from o-respose. I the sequece of estmatg the populato mea of study varable, we propose two dfferet types of famles of estmators for populato mea stratfed radom samplg usg a auxlary varable over the stuato whch o-respose s observed o both study ad auxlary varables. 3. eparate-type Famly of Estmators Followg gh ad hula 8, we ow propose a separate-type famly of estmators of populato mea X 0 stratfed radom samplg uder o-respose as where, T Fα= T Fα=T 0 p T Fα 5 AX f BT A f BX T, 6 A=α α, B=α α 4, =α α 3α 4 for α > 0ad f = /N. X s the populato mea of auxlary varable for the th stratum. The above famly ca geerate a umber of separate-type estmators of populato mea X 0 uder o-respose for the dfferet choces ofα. Partcularly, f we tae α =,, 3ad4, we get c 06 NP

3 J. tat. Appl. Pro. Lett. 3, No., / 9 ase If α = the A=B=0 ad = 6, so that TF = T X, 0 T X T hece TF = p T0 whch s usual separate rato estmator uder o-respose. ase If α = the A= = 0ad B=, so that TF = T T 0, X T therefore TF = p T0 whch s usual separate product estmator uder o-respose. X ase If α = 3the A=, B= ad =0, so that T F = T X f T 0, fx hece T F 3= p T X f T 0 fx whch s separate dual to rato-type estmator uder o-respose. The dual to rato-type estmator was troduced by rveataramaa 9. asev If α = 4the A=6, B=0 ad = 0, so that TF = T 0, cosequetly TF 4= p T0 = T 0st whch s usual mea estmator defed equato I order to obta the bas ad mea square error ME of the proposed famly, we use large sample approxmato. Let T0 = X 0e 0 ad T = X e such that Ee 0 =Ee =0, E e 0 = X N 0 L W 0, 0 E e = N L W ad X Ee 0 e = X 0 X N ρ 0 0 L W ρ 0 0 where X 0 s the populato mea of study varable for the th stratum. ρ 0 ad ρ 0 are the correlato coeffcets betwee study ad auxlary varables of etre group ad o-respose group respectvely for the th stratum. Expressg equato 5 terms of e 0 ad e, we get T F α = X 0 ad T F α X 0 = e 0 e φα e 0 e φα p X 0 e 0 e φα e 0 e φα A f B e φα... A f B e φα... where φα= f B A f B. Tag expectato both the sdes of the equato8 ad eglectg the terms of e 0, e havg power greater tha two, we get E TF α X 0 = p X 0 A f B φαe e Thus the bas of TF α up to the frst order of approxmatos gve by BTFα = φα p X 0 N A f B ρ 0 0 L p W X where 0 = 0 X 0, = X ad R 0 = X 0 X. A f B R 0 ρ 0 0 φαee0 e c 06 NP

4 0 M. K. haudhary, aurabh: Estmators for populato mea stratfed... Now the ME of TF α ca be obtaed as MTF α = M p TF = α = p E T F α X 0. p MT F α Puttg the value of TF α from equato7 to the above expresso, we get MT Fα = T F p E X 0 e 0 e φα e 0 e φα A f B e φα.. 0 X 0. Expadg the above expresso ad eglectg the terms of e 0, e havg power greater tha two, we get the ME of α up to the frst order of approxmato as MT F α = N p 0 φ αr 0 φαr 0ρ 0 0 L W p 0 φ αr 0 φαr 0ρ Optmum hoce of α I order to choose the optmum value ofα, we dfferetate M TF α wth respect to α ad equate the dervatve to zero. M TF α = α p φαφ αr 0 N φ αr 0 ρ 0 0 L W p φαφ αr 0 φ αr 0 ρ 0 0 = 0 3 where φ α s the frst dervatve of φα wth respect to α. From the above equato, we get { φα = p R 0 ρ 0 0 N { p N R0 } L W p R 0 ρ 0 0 / } L W p R 0 = ostat. 4 The above equatos a cubc equato α ad for a gve value of t provdes three real roots of αat whch the ME of TF α would attats mmum. I order to obta the optmum value of α amog the three real roots, the bas s taeto cosderato. We compute the bas of the estmator at the three real roots separately ad select the optmum value of α at whch bas s the least. 3.. Proposed Famly uder Dfferet Allocatos It s a well ow fact that the Neyma allocatos always preferable over the proportoal allocatof the strata mea squares are ow ad there s o o-respose the populato. But the presece of o-respose, haudhary et al. 6 have show that the Neyma allocatos ot always a better proposto eve the strata mea squares are ow. Wth ths bacgroud, they proposed some ew allocato schemes whch utlze the owledge of respose ad/or o-respose rates the case of o-respose. Moreover, t s observed that the presece of o-respose, the c 06 NP

5 J. tat. Appl. Pro. Lett. 3, No., / owledge of respose ad o-respose rates for dfferet strata ca easly be obtaed tha the strata varabltes from the past records or expereces. Now, we cosder the proposed famly of estmators uder proportoal allocato PA, Neyma allocato NA ad some of the ew allocato schemes proposed by haudhary et al. 6. Uder the proportoal ad Neyma allocatos, we have the sample sze for the th stratum = p for,,..., ad = p 0 / p 0 for,,..., respectvely. The M T F α uder the proportoal ad Neyma allocatos are respectvely gve by ad MTF α PA = N p 0 φ αr 0 φαr 0ρ 0 0 MT Fα NA = p 0 N L W p 0 φ αr 0 φαr 0 ρ 0 0 p 0 p 0 { p φ αr 0 } φαr 0 ρ 0 0 N 0 L p 0 W p 0 φ αr 0 φαr 0 ρ Uder the ew allocato scheme NA, f = p W / p W W s the respose rate the th stratum the we have MTFα NA = p W p N 0 φ αr 0 φαr 0 ρ 0 0 W p W L W p W 0 φ αr 0 φαr 0ρ Uder the ew allocato scheme NA, f = MT Fα NA = p W Uder the ew allocato scheme 3 NA3, f = MT F α NA3 = p W W p W p W the we get W p W p N 0 φ αr 0 φαr 0 ρ 0 0 L W p 0 φ αr 0 φαr 0ρ W W p W W p W W p W W N the we have p 0 φ αr 0 φαr 0ρ L W W p 0 φ αr 0 φαr 0 ρ ombed-type Famly of Estmators I the smlar maer, the combed-type famly of estmators of populato mea X 0 stratfed radom samplg whle both study ad auxlary varables go through the o-respose, s gve as T F α=t 0st AX f BT st A f BX T st. 0 c 06 NP

6 M. K. haudhary, aurabh: Estmators for populato mea stratfed... The above proposed famly may produce a umber of estmators of populato mea X 0 uder o-respose. For example fα =, we get TF = T X whch s usual combed rato estmator stratfed radom samplg whe both 0st T st the varables are subjected to o-respose. mlarly, for α =,3 ad 4, we have T F = T X f T st 0st T st X usual combed product estmator uder o-respose, TF 3 = T 0st combed dual to rato-type estmator uder o-respose ad fx TF 4=T 0st = T F 4usual mea estmator stratfed radom samplg uder o-respose respectvely. To obta bas ad ME oftf α, we use large sample approxmato. Let T0st = X 0e ad Tst = X e 3 such that Ee =Ee 3 =0, E e = X 0 E e 3 = X N N p 0 p L W p 0, L W p ad Ee 0 e = X 0 X N p ρ 0 0 L W p ρ 0 0. Uder the above assumptos, equato0 ca be expressed the terms of e ad e 3 as T Fα X 0 = X 0 e e φα e e 3 φα A f B e 3φα.... Tag expectato both the sdes of equato ad eglectg the terms of e ad e 3 havg power greater tha two, we get bas of TF α up to the frst order of approxmato E TFα X 0 = X 0 A f B φαe e 3 φαee e 3 BT Fα= φα X L W p p N A f B R 0 ρ 0 0 A f B R 0 ρ 0 0 where R 0 = X 0. X I the sequece of obtag the ME of TF α up to the frst order of approxmato, we square both the sdes of equato ad tae expectatogorg the terms of e ad e 3 havg power greater tha two E TFα X 0 = X 0 E e φ αe e 3 φαee e 3 MT F α= N p 0 φ αr 0 φαr 0ρ 0 0 L W p 0 φ αr 0 φαr 0 ρ Optmum hoce of α To obta the mmum ME oft F α, we fd the optmum value of α o dfferetatg the M T F α wth respect to αad equatg the dervatve to zero. Thus we have c 06 NP

7 J. tat. Appl. Pro. Lett. 3, No., / 3 M T F α α = N L W p φα = N p ρ 0 0 N p R 0 p φαφ αr 0 φ αr 0 ρ 0 0 φαφ αr 0 φ αr 0 ρ 0 0 = 0 L W p ρ 0 0 L W p R 0 = ostat. 4 ce φα s a cubc fucto of the parameter α. Thus for a gve value of, equato 4 provdes three real roots of αat whch we get the mmum ME of T F α. 3.. Proposed Famly uder Dfferet Allocatos I ths secto, we dscuss the proposed combed-type famly of estmators uder proporto allocato, Neyma allocato ad some ew allocato schemes cosdered secto 3... The sample sze of th stratum ad ME of the proposed famly uder proportoal allocato are respectvely gve by = p for,,..., ad MTF α PA = N p 0 φ αr 0 φαr 0ρ 0 0 L W p 0 φ αr 0 φαr 0 ρ Uder Neyma allocato, the sample sze of th stratum ad ME of TF α are respectvely gve by = p 0 for,,..., ad p 0 MT F α NA = p 0 N p 0 N 0 p 0 p { φ αr 0 φαr 0ρ 0 0 } by by L p 0 W p 0 φ αr 0 φαr 0ρ The sample sze of th stratum ad ME of TF α uder the ew allocato scheme NA are respectvely gve = p W p W for,,..., ad MT F α NA = p W p N 0 φ αr 0 φαr 0ρ 0 0 W p W L W p W 0 φ αr 0 φαr 0ρ Uder the ew allocato scheme NA, the sample sze of th stratum ad ME of TF α are respectvely gve c 06 NP

8 4 M. K. haudhary, aurabh: Estmators for populato mea stratfed... = p W p W for,,..., ad MT Fα NA = W p W p N 0 φ αr 0 φαr 0 ρ 0 0 by p W L W p 0 φ αr 0 φαr 0ρ The sample sze of th stratum ad ME of TF α uder the ew allocato scheme 3 NA3 are respectvely gve = p W W p W W for,,..., ad MT F α NA3 = W W p W W N p 0 φ αr 0 φαr 0ρ 0 0 p W W L W W p 0 φ αr 0 φαr 0ρ Emprcal tudy I order to support the theoretcal results, we have tae the data cosdered by haudhary et al. 7. There are 84 mucpaltes dvded to four strata cosstg of 73, 70, 97 ad 44 mucpaltes respectvely. The populato the year 985 has bee cosdered as study varable whereas the populato the year 975 s assumed to be auxlary varable. The parameters of the populato are gve below: Table : Partculars of Data tratum No. N X 0 X 0 0 ρ 0 ρ c 06 NP

9 J. tat. Appl. Pro. Lett. 3, No., / 5 Table gves the values of φα for dfferet sets of o-respose rates W o whch TF α provdes the optmum estmates uder the dfferet allocato schemes. Table : φα for TF α uder Dfferet Allocato chemes L = φα uder tratum No. W % PA NA NA NA NA Table 3 represets the ME comparso of the proposed famly T F α at α = 4, α = ad α optuder dfferet allocato schemes for the dfferet sets of W. t. No. W % Table 3: ME of TF α uder Dfferet Allocato chemes L = MTF α uder PA NA NA NA NA3 α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt c 06 NP

10 6 M. K. haudhary, aurabh: Estmators for populato mea stratfed... Table 4 reveals the values of φα for the dfferet sets of W o whch the proposed famly TF α gves the optmum estmates uder the dfferet allocato schemes. Table 4: φα for TF α uder Dfferet Allocato schemes L = φα uder tratum No. W % PA NA NA NA NA Table 5 depcts the ME of T F α at α = 4, α = ad α opt uder dfferet allocato schemes for the choces of dfferet sets of W. t. No W % Table 5: ME of TF α uder Dfferet Allocato chemes L = MTF α uder PA NA NA NA NA3 α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt α = 4 α = α opt c 06 NP

11 J. tat. Appl. Pro. Lett. 3, No., / ocluso I the preset artcle, we have proposed some famles of factor-type estmators of populato mea stratfed radom samplg usg a auxlary varable uder o-respose. We have suggested separate ad combed-type famles of estmators of populato mea wheever o-respose s observed o both study ad auxlary varables. The optmum propertes of the suggested famles have bee coferred. The theoretcal study of the suggested famles has bee carred out uder the proportoal, Neyma ad some ew allocato schemes based o respose or/ad o-respose rates. I order to susta the theoretcal results, a emprcal study has also bee doe. The tables 3 ad 5 preset a salet feature of comparso of proportoal ad Neyma allocatos wth some ew allocato schemes based o respose or/ad o-respose rates through the suggested famles T F α ad T F α respectvely. I both the tables, the optmum estmators provde better estmates tha the usual separate combed rato ad mea estmators uder o-respose. It s also revealed that most of the stuatos for dfferet choces of W, the allocato schemes NA, NA ad NA3, depedg upo the owledge of p ad W or/adw, provde more prcsed estmates as compared to proportoal allocato as well as Neyma allocato. Refereces M. H. Hase ad, W. N.Hurwtz, The problem of o-respose sample surveys, Joural of the Amerca tatstcal Assocato, 4, B. B. Khare, Allocato stratfed samplg presece of o-respose, Metro, 45 I/II, M. G. M. Kha, E. A. Kha ad M. J. Ahsa, Optmum allocato multvarate stratfed samplg presece of o-respose, J. Id. oc. Agrl. tatst., 6, M. K. haudhary, R. gh, R. K. hula ad M. Kumar, A famly of estmators for estmatg populato mea stratfed samplg uder o-respose, Pasta Joural of tatstcs ad Operato Research. Vol. 5-No., M. Khoshevsa, R. gh, P. hauha, N. awa ad F.maradache, A geeral famly of estmators for estmatg populato mea usg ow value of some populato parameters, Far East Joural of Theoretcal tatstcs,, M. K. haudhary, V. K. gh, R. gh ad F. maradache, O some ew allocato schemes stratfed radom samplg uder o-respose, tudes amplg Techques ad Tme seres, Zp Publshg 33 hesapeae Aveue olumbus, Oho 43, UA, M. K. haudhary, V. K. gh ad R. K. hula, ombed-type famly of estmators of populato mea stratfed radom samplg uder o-respose, Joural of Relablty ad tatstcal tudes JR, Vol. 5, No., V. K. gh ad D. hula, Oe parameter famly of factor-type rato estmators, Metro, 45 -, T. rveataramaa, A dual to rato estmator sample surveys, Bometra, 67, Maoj K. haudhary s Professor Assstat the Departmet of tatstcs, Baaras Hdu Uversty, Varaas, INDIA. He has bee worg the feld of samplg theory ad methods of estmato. He has publshed more tha 5 research papers atoal ad teratoal jourals. aurabh s research scholar the Departmet of tatstcs, Baaras Hdu Uversty, Varaas, INDIA. Hs area of research s samplg theory. He has publshed more tha 5 research papers atoal ad teratoal jourals. c 06 NP