Multistage Median Ranked Set Sampling for Estimating the Population Median

Transcription

1 Jounal of Mathematcs and Statstcs 3 (: ISSN Scence Publcatons Multstage Medan Ranked Set Samplng fo Estmatng the Populaton Medan Abdul Azz Jeman Ame Al-Oma and Kamaulzaman Ibahm School of Mathematcal Scences Faculty of Scence and Technology Unvesty Kebangsaan Malaysa UKM Bang Selango Malaysa Abstact: We modfy RSS to come up wth new samplng method namely Multstage Medan Ranked Set Samplng (. The was suggested fo estmatng the populaton medan and to ncease the effcency of the estmato fo specfc value of the sample sze. The was compaed to the Smple Random Samplng ( Ranked Set Samplng (RSS and Medan Ranked Set Samplng (MRSS methods. It s found that gves an unbased estmate of the populaton medan of symmetc dstbutons and t s moe effcent than RSS and MRSS based on the same numbe of measued unts. Also t was found that the effcency of nceases n ( s the numbe of stage fo specfc value of the sample sze. Fo asymmetc dstbutons consdeed n ths study has a small bas close to zeo as nceases especally wth odd sample sze. A set of eal data was used to llustate the method. Keywods: Smple andom samplng; anked set samplng; medan anked set samplng; multstage anked set samplng; multstage medan anked set samplng INTRODUCTION Many samplng methods ae suggested n the lteatue fo estmatng the populaton paametes. In some stuatons whee the expemental o samplng unts n a study can be easly anked than quantfed McIntye [] poposed the sample mean based on RSS as an estmato of the populaton mean. He found that the estmato based on RSS s moe effcent than. Takahas and Wakmoto [] povded the necessay mathematcal theoy of RSS. Muttlak [3] suggested usng medan anked set samplng (MRSS to estmate the populaton mean. Al-Saleh and Al-Oma [4] suggested that the multstage anked set samplng (MS method to ncease the effcency when estmatng the populaton mean fo specfc value of the sample sze. Jeman and Al-Oma [56] poposed double pecentle anked set samplng (DPRSS and multstage medan anked set samplng ( methods espectvely fo estmatng the populaton mean. They found that DPRSS and ae moe effcent than the commonly used fo the same sample sze. Jeman et al. [7] suggested multstage exteme anked set samplng (MERSS method fo estmatng the populaton mean. In ths study ou objectves s to suggest fo estmatng the populaton medan and to compae the effcency of ths method wth RSS and MRSS. Samplng methods Ranked set samplng: To obtan a sample of sze m by the usual RSS as suggested by McIntye [] select m andom samples each of sze m fom the taget populaton and ank the unts wthn each sample wth espect to a vaable of nteest by vsual nspecton o any cost fee method. Fo measuement fom the fst sample the smallest ank unt s selected and fom the second sample the second smallest ank unt s selected. The pocess s contnued untl fom the mth sample the mth ank unt s selected. The method s epeated n tmes f needed to get a RSS of mn. Multstage medan anked set samplng: The pocedue s descbed as follows: Step : Randomly selected m + unts fom the taget populaton whee s the numbe of stages and m s the sample sze. Step : Allocate the m + selected unts as andomly as possble nto m sets each of sze m. Step 3: Fo each m sets n Step ( f the sample sze m s odd select fom each m sets m + /th smallest ank unt.e. the the ( Coespondng Autho: Ame Ibahm Al-Oma School of Mathematcal Scences Faculty of Scence and Technology Unvesty Kebangsaan Malaysa UKM Bang Selango Malaysa 58

2 J. Math. & Stat. 3 (: medan of each set. If the sample sze m s even select fom the fst m / sets the ( m / th smallest ank unt and fom the second m / sets the (( m + / th smallest ank unt. Ths step yelds m medan anked sets each of sze m. Step.4: Wthout dong any actual quantfcaton f the sample sze m s odd select fom each m sets the (( m + /th smallest ank unt.e. the medan of each set. If the sample sze m s even select fom the fst m / sets the ( m / th smallest ank unt and fom the second m / sets select the (( m + / th smallest ank unt. Ths step yelds m medan anked sets each of sze m.e. t s the second stage medan anked sets. Step 5: The pocess s contnued usng Steps (3 and (4 untl we end up wth one th stage medan anked set sample of sze m fom. The whole pocess can be epeated n tmes to obtan a sample of sze nm fom. It s necessay to note hee that the ankng at all stages ae done by vsual nspecton o by any othe cheap method and the actual quantfcaton s exactly done on the last sample of sze m that s obtaned at the last stage. To estmate a populaton medan by a sample of sze m usng method we only andomly select m unts and fnd the medan. And f we use the RSS method fo the same estmaton we have to dentfy m unts and measue only m of them. But when we use we andomly select m + unts and measue only m of them. In each method RSS o we andomly select dffeent numbe of unts but measue only the same numbe of unts fo compason. Theefoe the measued unts of whch ae based on m + unt have moe nfomaton and moe epesentatve of the taget populaton when compaed to o RSS. Let us consde the followng example to llustate the method. Example : Consde the case of m = 3 and = 3. Theefoe we have a andom sample of sze m + = 8 unts as follows: Let ( j: m be the jth mnmum ( j = 3 of the th set ( =...7 at stage ( = 3. Allocate the 8 selected unts nto 7 sets each of sze 3 at zeo stage (. Fo = ank the unts wthn each sample vsually accodng to the vaable of nteest. Afte ankng the sets appea as shown below: A ( { ( ( ( = (:3 (:3 (3:3 } ( =...7. Now select the medan fom the 7 sets fo m = 3 the medan s the second smallest ank unt so that let ( ( ( :3 = med ( A ( =...7. Ths step yelds 7 medans whch ae ( :3 ( (... :3 7( :3. Allocate them nto 9 sets each of sze 3 as: A ( { ( ( ( = :3 ( :3 ( :3 } ( =...9. Fo = ank the unts wthn the 9 sets yelds fom the fst stage to get A ( { ( ( ( = :3 ( :3 ( 3:3 } ( =...9 and then select the medan fom each set as: ( ( ( :3 = med ( A ( =...9. ( ( ( Ths step yelds 9 medans ( :3 ( :3... 9:3 whch ae allocated nto 3 sets of medans each of sze 3 as: A ( { ( ( ( = :3 ( :3 ( :3 } ( = 3. Fo = 3 ank the unts wthn each set yelds at stage to obtan A (3 { (3 (3 (3 = :3 ( :3 ( 3:3 } ( = 3. Now select the medan of the thee sets as: (3 (3 = A ( = 3. med :3 ( { } (3 (3 (3 Ths step yelds :3 ( :3 3( :3 to be thd stage medan anked set sample. The actual quantfcaton fo estmatng the populaton medan of the vaable of nteest can be acheved usng only these thee unts. It s clea that the numbe of quantfed unts whch s 3 s a small poton of 7 sampled unts. Example : Consde the case of m = 4 and = 3 so that we have a andom sample of sze m + = 56 unts whch ae: Allocate the 56 unts nto 64 sets each of sze 4. Fo = ank the unts wthn each set wth espect to the vaable of nteest as follows: ( ( ( ( ( A = ( = { (:4 (:4 (3:4 (4:4} 59

3 Now select the second ank unt fom the fst 3 sets and the thd ank unt fom the othe 3 sets as: ( ( (:4 = nd mn ( A ( =...3 and ( ( (3:4 = 3d mn ( A ( = ( ( Ths step yelds 64 unts whch ae J. Math. & Stat. 3 (: (:4 (:4 ( ( ( (... 3(:4 33(3:4 34(3: (3:4. Allocate them nto 6 sets each of sze 4 as follows: A ( = ( ( ( ( ( =...8 and { (:4 (:4 (:4 (:4} { (3:4 (3:4 (3:4 (3:4} A ( ( ( ( ( = ( = Fo = ank the unts wthn each the 6 sets yelds fom the fst stage as: A ( = ( ( ( ( ( =...6. { (:4 (:4 (3:4 (4:4} Now fom the fst 8 sets select the second ank unt and fom the second 8 sets the thd ank unt as shown below: ( ( (:4 = nd mn A ( =...8 ( ( A ( ( (3:4 = 3d mn ( = ( ( ( ( Ths step yelds (:4 (:4... 8(:4 9(3:4 ( ( 0(3:4... 6(3:4. Allocate these unts nto 4 sets each of sze 4 as follows: A ( = ( ( ( ( ( = and { (:4 (:4 (:4 (:4} { (3:4 (3:4 (3:4 (3:4} ( ( ( ( ( A = ( = 34. Fo = 3 ank the unts wthn the last 4 sets yelds fom the second stage as: A (3 = (3 (3 (3 (3 ( = 34. { (:4 (:4 (3:4 (4:4} Now fom the fst sets select the second ank unt and fom the second sets the thd ank unt as shown below: (3 ( (:4 = nd mn A ( = and ( (3 (3 (3:4 = 3nd mn ( A ( = 34. The fnal set { (3 (3 (3 (3 (:4 (:4 3(3:4 4(3:4 } s a thd stage medan anked set of sze 4. RESULTS AND DISCUSSION Let... m be a andom sample wth pdf f( x cdf F( x a fnte mean µ and vaance σ. Let m ; m ; ; m m.. mm be ndependent andom vaables all wth the same dstbuton functon F( x. Let ( m : denotes the 60 th ode statstc of a sample of sze m (... m =. The estmato of the populaton medan fom a sample of sze m s defned as: ˆ η = + m f m s odd f m s even ( to be the mddle o the aveage of the two mddle unts afte sotng. The estmato of the populaton medan η fo a RSS of sze m s gven by: ˆ η RSS = medan { (: =... m}. ( If the sample sze m s odd let be the medan of the th sample (... m m + : m measued unts m + = at stage. The : m ae d and denote the measued O. If the sample ( sze m s even let m / th smallest m be the ank unt of the th sample (... m/ be the (( / th m + = and m + smallest ank unt of the th sample ( = ( m+ / ( m+ 4 /... m at stage. Note that the fst m / unts whch ae m : m whch ae m m : m m + m + m : m ae d and the second m / unts... m m m m m m : m : m : m : m ae d. Howeve + + ( whch denote the measued E ae ndependent but not dentcally dstbuted. The estmato of the populaton medan usng O n the case of an odd sample sze can be defned as: ( ˆ ηo = medan... ( (3 : m and fo an even sample sze the E estmato s defned as:

4 J. Math. & Stat. 3 (: Table : The effcency of RSS and elatve to fo estmatng the populaton medan of some symmetc dstbutons wth m = 3 45 fo = 34 Dstbutons η m RSS = = =3 =4 Unfom ( Nomal ( Nomal ( Logstc ( Table : The effcency of RSS and O elatve to fo estmatng the populaton medan of some asymmetc dstbutons wth m = 3 fo = 34 Dstbutons η RSS = = =3 =4 Exponental ( Eff Bas Log Nomal (0.000 Eff Bas Webull (3.079 Eff Bas Beta ( Eff Bas Gamma (3.674 Eff Bas Table 3: The effcency of RSS and E elatve to fo estmatng the populaton medan of some asymmetc dstbutons wth m = 4 fo = 34 Dstbutons η RSS = = =3 =4 Exponental ( Eff Bas Log Nomal (0.000 Eff Bas Webull (3.079 Eff Bas Beta ( Eff Bas Gamma (3.674 Eff Bas m m m : m : m ˆ ηe = medan... m+ :. (4 m : m The estmato of the populaton medan has the followng popetes:. If the dstbuton s symmetc about the populaton mean µ then fo any stage we have: a. ˆ η s an unbased estmato of a populaton medan η. Va ˆ η < Va( ˆ η. b. ( c. ( ˆ η Va < Va( ˆ ηrss.. If the dstbuton s asymmetc then 6

5 J. Math. & Stat. 3 (: Table 4: The effcency of RSS and O elatve to fo estmatng the populaton medan of some asymmetc dstbutons wth m = 5 fo = 34 Dstbutons η RSS = = =3 =4 Exponental ( Eff Bas Log Nomal (0.000 Eff Bas Webull (3.079 Eff Bas Beta ( Eff Bas Gamma (3.674 Eff Bas Table 5: The effcency of RSS and E elatve to fo estmatng the populaton medan of asymmetc dstbutons wth m = 6 fo = 34 Dstbutons η RSS = = =3 =4 Exponental ( Eff Bas Log Nomal (0.000 Eff Bas Webull (3.079 Eff Bas Beta ( Eff Bas Gamma (3.674 Eff Bas Table 6: The effcency of O wth espect to fo estmatng medan olve yelds pe tee fo m = 3 and = 34 O Effcency Medan Bas MSE Medan Bas MSE a. ˆ η s a based estmato of the populaton medan and the bas s vey small. In the case of odd sample sze ths bas s close to zeo as nceases. ( b. MSE( ˆ η MSE( ˆ < η. ( c. MSE( ˆ η MSE( ˆ < ηrss. 3. The effcency of ˆ η s nceasng n fo both dstbutons ethe symmetc o asymmetc about the populaton mean µ. Smulaton study based on fo medan estmaton: In ths secton we shall compae the poposed estmatos fo the populaton medan usng wth RSS and MRSS methods. Seveal pobablty dstbuton functons ae consdeed: unfom nomal logstc exponental lognomal webull beta and gamma. The effcency of ˆRSS η and ˆ η elatve to ˆ η f the dstbuton s symmetc s defned as: Va ( ˆ η eff ( ˆ η ˆ ηrss = (5 Va ˆ η eff ( ˆ η ˆ η ( RSS ( ˆ η Va = (6 Va ( ˆ η espectvely and f the dstbuton s asymmetc the effcency espectvely s gven by MSE ( ˆ η eff ( ˆ η ˆ ηrss = (7 MSE( ˆ ηrss 6

6 J. Math. & Stat. 3 (: ( ˆ η MSE eff ( ˆ η ˆ η =. (8 MSE( ˆ η The effcency values fo symmetc dstbutons whch ae unfom nomal and logstc fo estmatng the populaton medan wth m = ae pesented n Table. Whle fo asymmetc dstbutons whch ae exponental lognomal webull beta and gamma the effcency and the bas values fo estmatng the populaton medan usng wth m = and = 3 4 ae pesented n Tables -5 espectvely. Based on Tables -5 we can conclude the followngs:. If the undelyng dstbuton s symmetc we have: a. A gan n effcency s obtaned by usng fo dffeent values of m. ˆ b. η s an unbased of the populaton medan η. c. The effcency of ˆ η ( elatve to ˆ η s nceasng n. Fo example wth m = 5 the effcency of fo estmatng the medan of the logstc dstbuton fo = 3 and 4 espectvely ae and d. ˆ η s moe effcent than ˆRSS η fo all sample szes consdeed n ths study. e. Fo = the s same as MRSS. It s found that s moe effcent than MRSS fo.. If the undelyng dstbuton s asymmetc then we have the followngs: a. A gan n effcency s obtaned usng fo estmatng the populaton medan. ˆ b. The effcency of η s nceasng n fo specfc value of the sample sze. c. In the case of odd sample sze ˆ η has a small bas whch appoaches to zeo as nceases. Fo example fo m = 5 the effcency of fo estmatng the populaton medan of an exponental dstbuton when = s 3.96 wth bas 0.06 whle when = 4 the effcency s 8.90 wth bas d. In the case of even sample sze although thee s no clea patten of the bas value fo ˆ η but ths value s small and geneally close to zeo fo all asymmetc dstbutons consdeed. e. It s found that s moe effcent than MRSS. As an example fo estmatng the medan of the standad exponental dstbuton wth m = 5 the effcency of MRSS 3.96 wth bas 0.06 whle the effcency s 8.90 wth bas 0.00 fo = 4. Applcaton to eal data set: We llustate the pefomance of method fo medan estmaton usng a set of eal data consstng of the olve yelds of 64 tees. All samplng was done wthout eplacement. We obtan the medan and the MSE of each sample usng and method wth sample sze m = 3. We compaed the aveages of the sample estmates. Let u be the olve yeld of the th tee = 64. The mean µ and the vaance σ of the populaton espectvely ae 64 µ = u = kg/tee 64 = and σ ( µ 6. kg / tee 64 = u = 64 = The coeffcent of skewness and medan of the populaton ae and 8.50 espectvely. It s known that the coeffcent of skewness s zeo fo symmetc dstbuton but fo ou data the coeffcent s ndcatng that these data ae asymmetcally dstbuted. Fo llustaton we consde m = 3 fo = 3 4. The effcency of O elatve to ae computed usng Equaton (8 and ae pesented n Table 6 along wth the assocated bas. It can be seen fom Table 6 that the medans based on O ae much close to the populaton medan when compaed to those obtaned usng. It s also found that the effcency of O nceases n but the bas deceases n. CONCLUSION It can be concluded that s moe effcent than RSS and MRSS methods n estmatng the populaton medan based on the same sample sze. Also estmato of the populaton medan obtaned by method s an unbased when the undelyng dstbuton s symmetc about ts mean. If the undelyng dstbuton s asymmetc the estmato s found to have a small bas. The s ecommended to be used fo estmatng the populaton medan fo symmetc dstbutons. Fo asymmetc dstbutons ths method s suggested fo odd sample sze as the bas deceases n. 63

7 J. Math. & Stat. 3 (: REFERENCES. McIntye G.A. 95. A method fo unbased selectve samplng usng anked sets. Austalan J. Agl. Res. 3: Takahas K. and K. Wakmoto 968. On unbased estmates of the populaton mean based on the sample statfed by means of odeng. Ann. Inst. Statst. Math. 0: Muttlak H.A Medan anked set samplng J. Appl. Statst. Sc. 6: Al-Saleh M.F. and A.I. Al-Oma 00. Multstage anked set samplng. J. Statst. Plann. Infeence 0: Jeman A.A. and A.I. Al-Oma 006. Double pecentle anked set samples fo estmatng the populaton mean. Adv. & Appl. n Stat. 6: Jeman A.A. and A.I. Al-Oma 006. Multstage medan anked set samples fo estmatng the populaton mean. Pak. J. Stat. : Jeman A.A. A.I. Al-Oma and K. Ibahm 007. Multstage exteme anked set samplng fo estmatng the populaton mean. To appea n J. Statst. Theoy & Appl. Specal Issue of JSTA Ranked Set Samplng. 64