On Unequal Probability Sampling Without Replacement Sample Size 2
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1 Int J Oen Problems Com Math, Vol, o, March 009 On Unequal Probablt Samlng Wthout Relacement Samle Sze aser A Alodat Deartment of Mathematcs, Irbd atonal Unverst, Jordan e-mal: n_odat@ahoocom Communcated b Esam Eldn A Rakha Abstract A new selecton rocedure has been develoed for use wth the Hurvts- Thomson estmator Some results have been verfed for frst and second order ncluson robabltes Kewords: Unequal robablt samlng, Hurvts-Thomson estmator, varance estmator Introducton The concet of samlng wth unequal robablt wthout relacement was frst ntroduced b Meadow ( Horvtz and Thomson ( were the frst to gve theoretcal frame work of unequal robablt samlng wthout relacement The estmaton roosed b HT ( was: ^ = s ( Where s robablt of ncluson of th unt n the samle The varance of Horvtz-Thomson estmator was: ^ V = =, =, wth an unbased varance estmator ( HT
2 09 aser A Alodat ^ n n v HT = = = = The exresson for varance of Hurvts-Tomson gven b Yates Grund (4 s: V (3 ( = < = ^ YG HT (4 Wth an unbased varance estmator: ^ n n vyg HT = (5 = < Shahbaz and Hanf (3 suggested a new rocedure where selectng a samle of sze two where frst unt selected b robablt roortonal to and ( second wth robablt roortonal to ew Selecton Procedure In ths secton, we have gven a new selecton rocedure for use wth the Hurvts- Thomson ( estmator wth samle sze a Select frst unt wth robablt roortonal to and wthout relacement b Select second unt wth robablt The frst ncluson robablt s = B B ( = ( B where, = ( ( ( ( B = = ( B ( ( = ( ( (
3 On Unequal Probablt Samlng 0 ( / ( / = B ( ( ( ( ( Some results for new selecton rocedure Results (: = = n Proof: summng both sde of ( = = = B = ( ( ( B B = ( ( = = = ( ( ( ( ( ( ( ( ( ( ( = B = B = = = 3 B B = Snce n=, therefore equaton (3 can be wrtten as Result (: = ( n Proof: Summng both sdes of (, we get: = n = = B ( ( ( ( B ( ( ( ( (
4 aser A Alodat ( B B ( ( = ( ( Comarng (4 wth (, t can be seen that ( ( ( ( ( ( = Result (3: = n( n =, = ( 4 Alng double summaton on both sde of ( = =, = B ( ( ( ( ( 5 = =, B ( ( ( ( also = (6 =, Substtutng (5 n (6 we get: = = (7 =, = = = =, Snce n=two therefore equaton (7 can be wrtten as: = n( n Result (4: = = The value of and reduces to the standard results of smle random samlng Puttng = = n (, we get: = B (8 B = = = = = (9
5 On Unequal Probablt Samlng From (8 and (9 we get = Ths s for smle random samlng wthout relacement for a samle of sze equal two For substtutng = = and the value of B, n ( we get: = = B ( Whch s the ont ncluson robablt for a samle of sze two n case of smle random samlng wthout relacement 3 Oen Problems In ths aer, we suggest the use of new rocedure for selecton a samle of sze two b unequal robablt wthout relacement The roblem consdered n ths aer can be extend for a samle of sze greater than two Moreover, there s needed to comare between roosed estmator and the extended on References [] Hurvtz, DGand Thomson, DJ (95A generalzaton of samlng wthout relacement from a fnte unverse, JAmerStatAssoc47, [] Madow, WG (949On the theor of sstematc samlng IIAnnMathStat0, [3] Shahbas, MQand Hanf, M A samle rocedure for unequal robablt samlng wthout relacement samle sze PakJStatst003 Vol9 (, 5-60 [4] Yates, Fand Grund, PM (953Selecton wthout relacement from wthn strata wth robablt roortonal to szejrostatsoc B,5,53-6
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