An Efficient Class of Estimators for the Finite Population Mean in Ranked Set Sampling

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1 Open Journal of Statitic Publihed Online June 06 in SciRe An Efficient Cla of Etimator for the Finite Population Mean in Ranked Set Sampling Lakhkar Khan Javid Shabbir Department of Statitic Government College Toru Khber Pukhtunkha Pakitan Department of Statitic Quaid-i-Azam Univerit Ilamabad Pakitan Received 4 Februar 06; accepted June 06; publihed 4 June 06 Copright 06 b author Scientific Reearch Publihing Inc Thi ork i licened under the Creative Common Attribution International Licene (CC BY Abtract In thi paper e propoe a cla of etimator for etimating the finite population mean of the tud variable under Ranked Set Sampling ( hen population mean of the auiliar variable i knon The bia Mean Squared Error (MSE of the propoed cla of etimator are obtained to firt degree of approimation It i identified that the propoed cla of etimator i more efficient a compared to [] etimator everal other etimator A imulation tud i carried out to judge the performance of the etimator Keord Ranked Set Sampling Auiliar Variable Bia Mean Squared Error Relative Efficienc Introduction The problem of etimation in the finite population mean ha been idel conidered b man author in different ampling deign In application there ma be a ituation hen the variable of interet cannot be meaured eail or i ver epenive to do o but it can be ranked eail at no cot or at ver little cot In vie of thi ituation [] introduced the Ranked Set Sampling ( procedure [3] proved the mathematical theor that the ample mean under a an unbiaed etimator of the finite population mean more precie than the ample mean etimator under imple rom ampling (SRS The auiliar information pla an important role in increaing efficienc of the etimator [4] uggeted an etimator for population ratio in hoed that it had le variance a compared to uual ratio etimator in imple rom ampling (SRS In perfect ranking of element a conidered b [] [3] for etimation of population mean In ome ituation ranking ma not be perfect According to [5] the ample mean in i an unbiaed etimator of the Ho to cite thi paper: Khan L Shabbir J (06 An Efficient Cla of Etimator for the Finite Population Mean in Ranked Set Sampling Open Journal of Statitic

2 L Khan J Shabbir population mean regardle of error in ranking of the element In [6] the ranking of element a done on bai of the auiliar variable intead of judgment [] uggeted an etimator for population mean ranking of the element a oberved on bai of the auiliar variable [7] had uggeted a cla of Hartle-Ro tpe unbiaed etimator in [8] had alo propoed unbiaed etimator in tratified ranked et ampling In thi paper e ugget a cla of etimator for the population mean uing knon population mean of the auiliar variable in It i hon that the propoed cla of etimator outperform a compared to the [9] [] everal other etimator Alo ome pecial cae of the propoed cla are conidered in Table A (Appendi Ranked Set Sampling Procedure In ranked et ampling ( e elect m rom ample each of ize m unit from the population rank the unit ithin each ample ith repect to a variable of interet In order to facilitate the ranking the deign parameter m i choen to be mall From the firt ample the unit having the loet rank i elected from the econd ample the unit having econd loet rank i elected the proce i continued until from the lat ample the unit having the highet rank i elected In thi a e obtain m meaured unit one from each ample The ccle ma be repeated r time until mr unit have been meaured Thee n mr unit form the data Suppoe that the variable of interet Y i difficult to meaure to rank but there i the auiliar variable X hich i correlated ith Y The variable X ma be ued to obtain the rank of Y To perform the ampling procedure m bivariate rom ample each of ize m unit are dran from the population then each ample i ranked ith repect to one of the variable Y or X Here e aume that the perfect ranking i done on bai of the auiliar variable X hile the ranking of Y i ith error An actual meaurement from the firt ample i then taken of the unit ith the mallet rank of X together ith the variable Y aociated ith the mallet rank of X From the econd ample of ize m the Y aociated ith the econd mallet rank of X i meaured The proce i continued until from the mth ample the Y aociated ith the highet rank of X i meaured The ccle i repeated r time until n mr bivariate unit have been meaured out of the total mr elected unit 3 Some Eiting Etimator Notation We conider a ituation hen rank the element on the auiliar variable Let [ ] i j ( i be the ith judgment j ordering in the ith et for the tud variable Y baed on the ith order tatitic of the ith et of the auiliar variable X at the jth ccle Baed on the ample mean etimator ( of the population mean ( Y i given b [ ] ( here [ ] ( mr j i [ i] j r m To obtain the bia MSE of etimator e define: uch that here Y e X e [ ] ( 0 ( E( e E e ( 0 γ E e C W 0 0 E e γ C W E ee γρcc W 0 m m m W τ W W [ ] i τ i τ i m rxy m rx m ry i i i ( τ µ i i X τ [ ] µ i i [ ] Y τ µ i i [ ] Y µ X C i ρcc C C are the 47

3 L Khan J Shabbir coefficient of variation of Y X repectivel It alo be noted that the value of µ i [ ] µ i are the mean of ith order tatitic from ome pecific ditribution (ee [0] The variance of under cheme i given b [4] propoed an etimator of the population ratio ( ( γ Var Y C W ( Y R under a: X ˆ [ ] R (3 ( When population mean ( X of the auiliar variable (X i knon the variable Y X are poitivel correlated [9] propoed the ratio etimator for population mean (Y baed on a The bia MSE of r r [ ] X (4 ( up to the firt degree of approimation are given b ( r γ ( ρ ( Bia Y C C C W W ( r γ ( ρ ( MSE Y C C C C W W W When population mean ( X of the auiliar variable (X i knon the variable Y X are negativel correlated then the product etimator baed on i defined a: The bia MSE of p ( [ ] X (7 p up to the firt degree of approimation are given b ( p ( γρ Bia Y C C W (8 ( p { γ ( ρ ( } MSE Y C C C C W W W (9 [] uggeted an etimator under i defined a: here λ i uitabl choen contant The minimum bia MSE of at optimum value of λ ie are given b λ Bia ( opt MSE λ [ ] (0 ( γ C W min min Y ( γ C W ( γ C W Y ( γ C W ( γ C W The difference-tpe etimator for population mean ( Y baed on i given b (5 (6 ( ( 48

4 L Khan J Shabbir [ ] d d ( X ( here d i a contant The minimum variance of d( at optimum value of d ie i given b d ( opt (3 ( γ ( γ C W R C W Var ( d Y γ C min W ( γ C W ( γ C W Folloing [] [] uggeted a cla of etimator of the population mean (Y baed on a: g ax b λ[ ] λ [ ] S α( a b ( α ( ax b here α i a uitabl choen contant a b are either real number or function of knon parameter of the auiliar variable X g i a calar hich take value of (for generating ratio-tpe etimator (for generating product-tpe etimator ( λ λ are contant hoe um need not be unit The bia of S( i given b g ( S λ λ λ α θ ( γ λ αθ γρ Y Bia Y ( g C W g C C W The MSE of S( to firt degree of approimation i given b here ( λ λ S λλ λ λ MSE Y A A B B C C D D ( γ C A A W {( } B γ C g g θ α C 4 gαθc B W g g θ α W 4 gαθw ( g g C C g C C γ θα θ α ( g g C W gθαc θ α W ( g g D γ θ α C gθαc ( We dicu to cae Cae : Sum of eight i unit (ie λ λ Solving (7 the optimum value of λ i given b g g D θ α W gθαw (4 (5 (6 (7 49

5 L Khan J Shabbir ( B B ( C C ( D D ( A A ( B B ( C C λ opt λ in (7 e get the minimum MSE of S( given b ( B B C C D D MSE ( Y ( ( S B B D D min ( A A ( B B ( C C Subtituting ( opt Cae : Sum of eight i fleible (ie λ λ Solving (7 the optimum value of λ λ are given b ( B B ( C C ( D D ( opt ( A A( B B ( C C λ λ ( A A ( D D ( C C ( opt ( A A( B B ( C C Subtituting the optimum value of λ λ in (7 e get MSE {( B B ( C C ( D D ( A A ( D D } { } ( Y S min ( A A( B B ( C C (8 (9 4 Propoed Cla of Etimator Folloing [] [] e propoe a cla of etimator of the population mean (Y under a ax b a b ax b k[ ] k( X α ep α L ( ax b ( a ( b ( a ( b here α i a uitabl choen contant a b are either real number or the function of knon parameter of the auiliar variable X ( k k are contant hoe um need not be unit From (0 e can generate a large number of etimator for the different value of the contant (Table A in Appendi The propoed etimator L( can be ritten in term of e 0 e a e 3 e ky ( e0 kxe θ θ α ( α( θe L ( 8 here θ ax ( ax b Solving ( e have α 5α Y Y( k ky θe kyθ e kye L 8 0 α α ky θee 0 kxe kx θe Taking epectation of both ide of above equation e get bia of L( given b 5α α Bia Y k kyθ L γc W kyθ C W 8 γ α kx θ ( γc W ( ( (0 ( (3 430

6 L Khan J Shabbir Squaring both ide of Equation ( ignoring higher order term of e e have α α ( Y Y ( k ky e0 θ e θee L 0 5α α kx e k( k Y θ e θe 8 α α k( k YX θe e kk YX θe e0e Taking epectation of both ide of above equation e obtain the MSE of L( a given b here ( ( ( L MSE Y k k E E k F F k k G G We dicu to cae Cae : Sum of eight i unit (ie k k The optimum value of k i given b k ( opt k k H H kk I I E Y C α α γ θ C θc α α E Y W θ W θw F X C F γ XW 5 G Y α α γ θ C θc 8 5α α G Y θ W θw 8 α H XY γ θc α H XY θ W α I XY C C γ θ α I XY θ W W i Y ( F F ( G G ( H H ( I I Y E E F F G G H H I I Thu the minimum MSE of L( i given b MSE { } { } E E Y H H F F I I G G ( L( min Y ( E E ( F F ( G G ( H H ( I I (4 (5 43

7 L Khan J Shabbir Cae : Sum of eight i fleible (ie k k For ( k k the MSE of L( in Equation (4 i minimized for { } { } { } { } F F Y G G H H H H I I opt ( F F Y ( G G ( E E H H I I k k { } { } ( H H {( E E ( G G } ( I I Y ( G G { } opt ( F F Y ( G G ( E E H H I I Subtituting the optimum value of k k in (4 e get ( L opt opt opt min MSE Y k k E E k F F ( ( opt ( opt ( k k G G k k H H ( opt ( opt opt opt k k I I Note: It i difficult to make the theoretical comparion due to compleit therefore e adopt the numerical tud 5 Simulation Stud We ue the ame data et a earlier ued b [] perform ome imulation tud to invetigate the performance of the etimator Population (ource: [3] Y Number of acre devoted to farm during 99 (ACRES9 X Number of large farm during 99 (LARGEF9 N 3059 ρ Y X 565 S 4538 S 73 We et r 0 m 5 to elect a ample of n mr 50 unit from the population of ize N 3059 To compute the value of W W W b imulation e eplain our imulation methodolog a follo Here W W W can be ritten a m W ( [ ] RDY i mr here W i m mr i mr i ( RDX ( i m W RDX i RDY i [ ] ( ( [ ] µ µ i [ ] i RDY i RDX ( i i m Y X To find the poible value of the ratio RDY [ i ] for m 5 e generate i ~ ( 0 [ ] e RDY [ ] e RDY [ 3] e3 RDY [ 4] 5 008e4 [ 5] e5 e N calculate RDY RDY It mean that hen the firt mallet value i elected from the ranked et ample the epected ratio of that value to the population mean could be cloe to 05 hen the econd mallet value i (6 43

8 L Khan J Shabbir elected the ratio of that value to the population mean could be cloe to 050 hen the third mallet value i elected the epected ratio of that value to the population mean ill cloe to Similarl the epected ratio of the fourth fifth value could be cloe to 5 75 repectivel In each cae e eighted error term e i ith a mall number 008 to make ure that the ratio RDY [ i ] remain poitive In other ord it mean that e are generating ei ~ N ( Thu the poible value of the ratio RDY [ i ] are epected to remain cloe to thoe e are conidering here Similarl for the poible value of the ratio RDX ( i e conider RDX ( e RDX e RDX ( e3 RDX ( e4 RDX ( e5 here ei ~ N ( 0 Here e eighted e i ith a mall number 005 becaue it ma be le rik to rank the auiliar variable X than the tud variable Y Thu the value of W i [ ] W i W are obtained through thi imulation are repreented in the lat three column of Table ( i Table PRE of propoed cla of etimator through imulation a b g α R ( 0 R ( 0 R ( 03 R ( 0 4 R ( 05 ( 06 R W W W i i ( i

9 L Khan J Shabbir We invetigate the percentage relative efficienc (PRE of ratio etimator ˆ r θ (a the Searl etimator ˆ θ the difference etimator ˆ d θ3 [] etimator ˆ S( θ4 hen λ λ ith repect to conventional etimator ˆ θ0 (a We alo calculate PRE of the propoed cla of etimator a ˆ θ L 5 hen ( k k hen ( k k a ˆ θ L 6 ith repect to ˆ θ0 The PRE of our propoed etimator other eiting etimator ˆj θ j 6 ith repect to conventional etimator ˆ θ0 i defined a PRE ( ˆ θ ˆ 0 θ j ( ˆ θ0 ( ˆ θ j MSE MSE 00 j 6 (7 The PRE of our propoed etimator other eiting etimator ith repect to conventional etimator are given in Table 6 Concluion Since abg α are the fied contant in [] etimator in the propoed cla of etimator There can be a large number of combination for different value of thee contant Here onl limited number of reult are reported in Table Obvioul it can be oberved through the imulation tud in Table that the propoed cla of etimator i more efficient than all conidered etimator It PRE increae from 645 to 78 hen α change from 0 to 09 but decreae lightl hen α i cloe to 05 Generall e can a PRE of propoed cla increae a value of α increae for fied value of contant a b g [] Cla of etimator ha maimum PRE 675 but it i le efficient a compared to the propoed cla of etimator for all the choice of contant reported in Table Alo from the Table e can ee that other competitor etimator are alo le efficient than the propoed cla of etimator If e make comparion beteen the to k k i more precie than the Cae propoed cae then the cla of etimator in Cae ( ( k k We can ee from Table that b fiing the value of a b at 5 the propoed clae of etimator give more precie reult hen the value of α i aa form 05 either cloe to 0 or While b fiing poitive value of the contant a b e get more precie reult for α cloe to 05 Therefore the propoed cla of etimator can be preferred over it competitive etimator in application under Acknoledgement The author ih to thank the editor the anonmou referee for their uggetion hich led to improvement in the earlier verion of the manucript Reference [] Singh HP Tailor R Singh S (04 General Procedure for Etimating the Population Mean Uing Ranked Set Sampling Journal of Statitical Computation Simulation [] Mclntre G (95 A Method for Unbiaed Selective Sampling Uing Ranked Set Crop Pature Science [3] Takahai K Wakimoto K (968 On Unbiaed Etimate of the Population Mean Baed on the Sample Stratified b Mean of Ordering Annal of the Intitute of Statitical Mathematic [4] Samai HM Muttlak MA (996 Etimation of Ratio Uing Ranked Set Sampling Biometrical Journal [5] Dell T Clutter J (97 Ranked Set Sampling Theor ith Order Statitic Background Biometric [6] Stoke SL (977 Ranked Set Sampling ith Concomitant Variable Communication in Statitic: Theor Method [7] Khan L Shabbir J (05 A Cla of Hartle-Ro Tpe Unbiaed Etimator for Population Mean Uing Ranked Set Sampling Hacettepe Journal of Mathematic Statitic 434

10 L Khan J Shabbir [8] Khan L Shabbir J (06 Hartle-Ro Tpe Unbiaed Etimator Uing Ranked Set Sampling Stratified Ranked Set Sampling North Carolina Journal of Mathematic Statitic 0- [9] Kadilar C Unazici Y Cingi H (009 Ratio Etimator for the Population Mean Uing Ranked Set Sampling Statitical Paper [0] Arnold BC Balakrihnan N Nagaraja HN (0 A Firt Coure in Order Statitic Vol 54 Siam [] Searl DT (964 The Utilization of a Knon Coefficient of Variation in the Etimation Procedure Journal of the American Statitical Aociation [] Khohnevian M Singh R Chauhan P Saan N Smarache F (007 A General Famil of Etimator for Etimating Population Mean Uing Knon Value of Some Population Parameter( Far Eat Journal of Theoretical Statitic 8-9 [3] Lohr S (999 Sampling: Deign Anali Dubur Pre Boton Appendi Table A Some pecial cae of the propoed cla of etimator k k α a b Etimator Remark ( [ ] Uual mean etimator X r( [ ] ( X λ λ r( [ ] ( ( β 0 0 reg( [ ] β X ( k k 0 0 d( [ ] k X X k 0 0 [ ] k dr ( X ( ( X k 0 0 k gdr( [ ] k ( X( ( X β 0 0 regr( [ ] β ( X ( ( X 0 0 [ ] ep e X X ( β 0 [ ] β ( X ep rege X ( ( ( Uual ratio etimaotr Kadilar et al (009 ratio tpe etimator Regreion tpe etimator Difference tpe etimator Difference-ratio etimator Generalied difference-ratio etimator Regreion-ratio etimator Eponential tpe etimator Regreion-eponential tpe etimator 435