On New Selection Procedures for Unequal Probability Sampling

Transcription

1 Int. J. Oen Problems Comt. Math., Vol. 4, o. 1, March 011 ISS ; Coyrght ICSRS Publcaton, On ew Selecton Procedures for Unequal Probablty Samlng Muhammad Qaser Shahbaz, Saman Shahbaz and Muhammad Hanf Deartment of Mathematcs, COMSATS Insttute of IT, Lahore, Pakstan e-mal: Deartment of Mathematcs, COMSATS Insttute of IT, Lahore, Pakstan e-mal: Deartment of Mathematcs, LUMS, Lahore Abstract We roose a new selecton rocedures n unequal robablty samlng that can be used wth Horvtz and Thomson Estmator. Exresson for ncluson robablty for th unt ( ) and ont ncluson robablty of th and th unt ( ) has been obtaned. Some desrable roertes of roosed rocedure. and has been verfed for Keywords: Horvtz and Thomson Estmator, Selecton Procedure, Unequal Probablty Samlng. 1 Introducton Unequal robablty samlng has ts roots n early thrtes. The remer dea of unequal robablty samlng was ntroduced by eyman [7] n hs ground breakng artcle. The frst mathematcal framework of unequal robablty samlng wth relacement was gven by Hansen and Hurwtz [5]. The technque roosed by Hansen and Hurwtz [5] could not fnd much alcablty due to ossblty of selecton of a oulaton unts more than once. Horvtz and Thomson [6] develoed general samlng theory by restrctng selecton of oulaton unts to samle for one tme only. The estmator of oulaton total roosed by Horvtz and Thomson [6] s gven as: yˆ y HT ; (1.1) S

2 Muhammad Shahbaz, Saman Shahbaz and Muhammad Hanf 178 where s robablty of selecton of th unt n the samle. The varance of Horvtz and Thomson [6] estmator has two dfferent forms. The varance of (1.1) roosed by Horvtz and Thomson [6] s gven as: 1 Var yˆ y y y HT ; (1.) where s ont robablty of ncluson of two unts n the samle. Another form of varance of (1.1) roosed by Sen [8] and ndeendently by Yates and Grundy [10] s: y y. (1.3) 1 1 ˆHT Var y The estmator of (1.3), roosed by Sen [8] and ndeendently by Yates and Grundy [10] s: var yˆ HT n n y y. (1.4) 1 1 The varance exresson gven n (1.3) s more oular as comared wth (1.). Both varance exressons gven n (1.) and (1.3) are based uon the quanttes and. Sutable choce of these quanttes can lead to substantal reducton n varance of Horvtz and Thomson [6] estmator. Survey statstcans, from tme to tme, has roosed number of selecton rocedures that can be used wth Horvtz and Thomson [6] estmator. These selecton rocedures have been roosed wth a vew that the varance of (1.1) s mnmum and (1.4) reman ostve for all ossble samles from a oulaton of sze. Brewer [] roosed a rocedure whch ensures that where s robablty of selecton of th unt n the samle. Shahbaz and Hanf [9] have roosed a more smler selecton rocedure for use wth (1.1). Brewer and Hanf [3] and Hanf and Brewer [4] has gven a comrehensve lst of selecton rocedures that can be used wth (1.1). Recently Alodat [1] has roosed a smler selecton rocedure that can be used wth Horvtz and Thomson [6] estmator by followng the lnes of Shahbaz and Hanf [9]. In the followng we roosed an extenson of the selecton rocedure roosed by Alodat [1]. The ew Selecton Procedure Suose a oulaton of unts s avalable and a samle of sze s to be selected. We roose the followng selecton rocedure for selecton of the samle:

3 179 On ew Selecton Procedures for Select frst unt wth robablty roortonal to q a wthout relacement. and Select second unt wth robablty roortonal to sze of remanng unts. The exresson for robablty of ncluson of th unt n the samle s derved below: q 1 q B a a a wth B B a 1 B a 1 B The exresson for ont ncluson robablty s: q q a 1 1 B 1 1 We now verfy some desrable roertes for (.1) and (.) n the followng. Result 1: The quantty satsfes Proof: Consder (.1) as: 1 n. a 1 B Alyng summaton on both sdes we have:.1.

4 Muhammad Shahbaz, Saman Shahbaz and Muhammad Hanf 180 a B B a B a 1 1 B a B B Snce n =, so from (.3) we can readly see that Result : The quantty satsfes n 1 Proof: Consder quantty from (..) as: 1 n. 1. a 1 1 B 1 1 Alyng condtonal summaton, we have: a a a B B 1 1 B a a B B a 1 B a 1 B Snce n = so n 1

5 181 On ew Selecton Procedures for Result 3: The quantty 1 1. satsfes nn1 Proof: The roof s straghtforward from roves of results 1 and. Result 4: The Sen Yates Grundy varance estmator s always non negatve under ths selecton rocedure. Proof: For non negatvty of Sen Yates Grundy varance estmator we must have 0. Usng (.1) and (.) we have: 1 1 a 1 1 a D B B D 1 D where D 1 1. The exresson (.5) s always non negatve and hence Sen Yates Grundy varance estmator s always non negatve under ths rocedure. Result 5: The quanttes smle random samlng for and 1 reduces to classcal results of Proof: The roof s straghtforward Oen Problem We have roosed a new selecton rocedure for use wth Horvtz and Thomson estmator. The rocedure can be extended to samle of any sze. The otmum value of constant a can be located by conductng emrcal study. References [1]. A. Alodat "On Unequal Probablty Samlng Wthout Relacement Samle Sze ", Int. J. Oen Problems Com. Math., Vol. (1), (009),

6 Muhammad Shahbaz, Saman Shahbaz and Muhammad Hanf 18 [] K.R. W. Brewer, A model of systematc samlng wth unequal robabltes Austral. J. Statstcs. 5, (1963), 5 13 [3] K. R. W. Brewer and M. Hanf. Samlng wth unequal robabltes Lecture notes n Statstcs seres, o. 15, Srnger Verlag, ew York (1983). [4] M. Hanf, and K. R. W. Brewer. On unequal robablty samlng wthout relacement; a revew Inter. Statst. Rev. 48(3), (1980), [5] M. H. Hansen and W.. Hurwtz. "On the theory of samlng from a fnte oulaton." Ann. Math. Stat. 14, (1943), [6] D. G. Horvtz and D. J. Thomson. A generalzaton of samlng wthout relacement from a fnte unverse J. Amer. Statst. Assoc. 47, (195), [7] J. eyman On the two d erent asects of the reresentatve method: the method of stratfeded samlng and the method of urosve selecton, J. Roy. Stat. Soc., 97, (1934), [8] A. R. Sen On the estmate of the varance n samlng wth varyng robabltes J. Indan Soc. Agr. Statst. 5, (1953), [9] M. Q. Shahbaz and M. Hanf A smle rocedure for unequal robablty samlng wthout relacement and a samle of sze, Pak. J. Stat., Vol. 19(1), (003), [10] F. Yates and P. M. Grundy Selecton wthout relacement from wthn strata wth robablty roortonal to sze J. Roy. Statst. Soc. B. 15, (1953),