A General Class of Selection Procedures and Modified Murthy Estimator

Transcription

1 ISS Journal of Statstcs Volume 4, 007,. 3-9 A General Class of Selecton Procedures and Modfed Murthy Estmator Abdul Bast and Muhammad Qasar Shahbaz Abstract A new selecton rocedure for unequal robablty samlng wthout relacement has been obtaned for samle sze "n". Some results regardng robablty of ncluson and ont robablty of ncluson have been verfed. A new estmator of oulaton total has been derved usng the dea of Murthy (957). Emrcal study has also been carred out. Keywords Unequal robablty samlng, Murthy estmator, Sen Mdzuno selecton rocedures. Introducton Unequal robablty samlng was frstly ntroduced n early fortes by Hansen and Hurwtz (943). The Horvtz and Thomson (95) were the frst to gve theoretcal framework of unequal robablty samlng wthout relacement. The estmator of oulaton total roosed by Horvtz and Thomson (95) s: Y y HT, (.) s where s robablty of ncluson of -th unt n the samle. Statstcal Offcer, State Bank of Pakstan, Karach Emal: bast_ravan97@hotmal.com Deartment of Mathematcs, COMSAT Insttute of Informaton Technology, Lahore Emal: qshahbaz@gmal.com

2 4 Bast and Shahbaz The varance of (.) obtaned by Sen (953) and ndeendently by Yates and Grundy (953) s gven as: ( ) Y Y V y HT (.) where s ont robablty of ncluson of -th and -th unt n the samle. Large number of selecton rocedures have been develoed for use wth (.). A comrehensve revew can be found n Brewer and Hanf (983). Murthy (957) roosed hs estmator of oulaton total and s gven as: n tsymm PS / y (.3) P S where PS s robablty of samle gven that -th unt s selected at the frst draw and PS s robablty of the samle. Murthy (957) estmator for a samle sze, under Yates Grundy (953) rocedure s: t y y symm wth (.4) P P P P Y Y Var t symm (.5) P P P P. The ew Selecton Procedure In ths secton we have gven a new selecton rocedure and we have also verfed some mortant results for the robablty of ncluson and ont robablty of ncluson.. The Procedure The new selecton rocedure s gven as under: Select frst unt wth robablty roortonal to and wthout relacement Select a random samle of sze (n-) from the remanng (-) unts.

3 A General Class of Selecton Procedure and Modfed Murthy Estmator 5 The robablty of ncluson for the -th unt n the samle for ths selecton rocedure s gven as: n (.) d d n n d d n n d d where d (.) The ont robablty of ncluson for -th and -th unts n the samle for ths selecton rocedure s gven as: n n n n d d n n n n (.3) d For the values of constant and, (.) and (.3) reduces to the exresson of robablty of ncluson and ont robablty of ncluson of Mdzuno (95) selecton rocedure.. Some Results For ew Selecton Procedure In ths secton we have verfed some of the common results for the quanttes and obtaned under the new selecton rocedure.

4 6 Bast and Shahbaz Result : The values of and reduces to the standard results of smle random samlng for. Result : n for ths selecton rocedure. Result 3: The quantty, obtaned under ths selecton rocedure, satsfes the. relaton n Result 4: The quantty, obtaned under ths selecton rocedure, satsfes the where n s the samle sze. relaton n n 3. Modfed Murthy Estmator In ths secton a new estmator of oulaton total wth unequal robablty s derved wth ts varance. The estmator has been derved by usng the new selecton rocedure n (.3). ow for new selecton rocedure, for a samle of sze, we have: PS / PS / and (3.) PS d Usng (3.) n (.3) the modfed Murthy estmator s obtaned as: t MM y y d t MM d y y (3.)

5 A General Class of Selecton Procedure and Modfed Murthy Estmator 7 The Estmator (3.) reduces to the estmator of smle random samlng for equal robabltes. The estmator (3.) s an unbased estmator of oulaton total and t s slght modfcaton of actual Murthy (957) estmator. The estmator (3.) s a class of modfed Murthy estmators and dfferent estmators can be constructed for dfferent values of and. Also for and, the estmator (3.) transforms to the classcal rato estmator for a samle of sze. The varance of (3.) has been obtaned n the followng secton. 3. Desgned Based Varance of the Modfed Estmator The varance of the modfed Murthy estmator s gven as: Var tmm tmm PS Y Substtutng the values of t MM from (3.) and P(S) from (3.) and smlfyng we get: y y Var tmm y y y y (3.3) P S or y y Var tmm PS y y PS (3.4) The varance exresson gven n (3.4) transforms to the exresson of smle random samlng for equal robabltes. 4. Emrcal Study The emrcal study has been carred out for the selected 0 natural numbers beng the varous values of the constant and n the range of -5 to 5 wth an ncrement of.0. For ths emrcal study the varance gven n (3.4) has been calculated. After calculatng the varance we have assgned ranks to each varance on the bass of ts magntude. The ar of constants and whch roduces smallest varance has been assgned a rank of ; the second smallest varance has

6 8 Bast and Shahbaz been assgned a rank and so on. Ths rocedure s reeated for all the oulaton and fnally we have comuted the average rank for each ar of the constants and. These average ranks for varous ars have been gven n the followng Table: Table: Average Ranks of the Varance of the modfed Murthy Estmator for the dfferent values of and α β Lookng at the Table we can readly see that for all the values of, the value of roduces the least average rank. We therefore ck as the sutable choce for ths constant. Further, from Table we can straght away decde that constant can be relaced wth and we can develo a sub-class of estmators for varous values of. Ths sub class has the form: t MM c y y (4.) where c

7 A General Class of Selecton Procedure and Modfed Murthy Estmator 9 Ths sub-class can be used by usng varous values of the constant n (4.) for estmaton of oulaton total. References. Brewer, K. R. W. and Hanf, M. (983). Samlng wth unequal robabltes. Lecture notes n Statstcs Seres, o. 5, Srnger Verlag, ew York.. Hansen, M. H. and Hurwtz, W.. (943). On the theory of samlng from a fnte oulaton. Annals of Mathematcal Statstcs, 4, Horvtz, D. G. and Thomson, D. J. (95). A generalzaton of samlng wthout relacement from a fnte unverse. Journal of Amercan Statstcal Assocaton, 47, Mdzuno, H. (95). On the samlng system wth robablty roortonate to sum of sze. Annals of Insttutonal Mathematcal Statstcs, 3, Murthy, M.. (957). Ordered and unordered estmators n samlng wthout relacement. Sankhya, 8, Sen, A. R. (953). On the estmate of the varance n samlng wth varyng robabltes. Journal of the Indan Socety of Agrcultural Statstcs, 5, Yates, F. and Grundy, P. M. (953). Selecton wthout relacement from wthn strata wth robablty roortonal to sze. Journal of Royal Statstcal Socety, Seres B, 5, 53 6.