REVISTA INVESTIGACION OPERACIONAL VOL. 8, NO., 66-71, 017 RATIO AND PRODUCT TYPE EXPONENTIAL ESTIMATORS FOR POPULATION MEAN USING RANKED SET SAMPLING Gajeda K. Vishwakaa 1*, Sayed Mohaed Zeesha *, Calos N. Bouza ** * Depatet of Applied Matheatics, Idia Istitute of Techology (ISM Dhabad Jhakhad-8600, INDIA. ** Depatet of Matheatics ad Coputatio, Uivesity of Havaa, Havaa, CUBA. ABSTRACT I this pape, we suggest a ipoved fo of expoetial atio ad poduct estiatos usig aked set saplig (RSS i ode to estiate the populatio ea Y of study vaiate Y usig auxiliay vaiate X. The ea-squaed eo (MSE of the poposed class of estiatos is obtaied. To judge the eits of the suggested estiatos ove othes a siulatio study has bee caied out. KEYWORDS: Raked set saplig, Auxiliay vaiate, Study vaiate, MSE, MSC: 6D0 RESUMEN E este tabajoo sugeios la foa ejoada expoecial de estiadoes de azó y poducto usado uesteo po cojutos odeados (RSS paa estia la edia poblacioal Y de ua vaiable de estudio Y usado la vaiable auxilia X. El eo cuadático edio (MSE de la clase de estiadoes popuesta es obteido. Paa evalua los éitos de los estiadoes sugeidos especto a otos se desaolló u estudio de siulació. 1. INTRODUCTION The aked set saplig (RSS is a ethod of saplig which povides oe stuctue to the collected saple ites ad iceases the aout of ifoatio peset i the saple. The ethod of aked set saplig (RSS was fist evisaged by McItye (19 as a cost-efficiet substitute to siple ado saplig (SRS fo those cicustaces whee easueets ae icoveiet o expesive to obtai but (judget akig of uits accodig to the vaiable of iteests, say, Y, is copaatively easy ad cheap. It is kow that the estiate of the populatio ea usig RSS is oe efficiet tha the oe obtaied usig SRS. McItye (19 ad Takahasi ad Wakioto (1968 cosideed pefect akig of the eleets, that is, thee ae o eos i akig the eleets. Yet, i ost cicustaces, the akig ay ot be doe pefectly. Dell ad Clutte (197 deostated that the ea usig the RSS is a ubiased estiato of the populatio ea, whethe o ot thee ae eos i akig. Stokes (1977 cosideed the case whee the akig is doe o the basis of a cocoitat (auxiliay vaiable X istead of judget. We would expect the vaiable of iteest will be highly coelated with the cocoitat (auxiliay vaiable. Stokes (1980 showed that the estiato of the vaiace based o RSS data is a asyptotically ubiased estiato of the populatio vaiace. Saawi ad Muttlak (1996 deal the poble of estiatig the populatio atio of the two vaiables Y ad X usig RSS pocedue. I additio, RSS has bee ivestigated by ay eseaches like Al-Saleh ad Al-Oay (00, Wolfe(00, Madowaa ad Mehta (01 Al-Oay ad Bouza (01. The RSS has ay statistical applicatios i agicultue, biology, evioetal sciece, edical sciece etc. Let ado saples of size bivaiate uits each ad ak the bivaiate uits withi each saple with espect to the auxiliay vaiate X. Next, select the i th sallest auxiliay vaiate X fo the i th saple fo i = 1,,, fo actual easueet of the associated vaiate of iteest Y with it. I this way, a total ube of easued bivaiate uits ae obtaied, oe fo each saple. The cycle ay be epeated ties to get a saple of size = bivaiate uits. These = uits built the RSS data. Note that we assue that the akig of the vaiate X will be pefect while the akig of the vaiate Y will be with eos, o at wost of a ado ode if the coelatio betwee Y ad X is close to zeo. Also, ote that i RSS, eleets ae idetified, but oly of the ae quatified. So, copaig this saple with a 1 vishwagk@ediffail.co, zeesha008x@gail.co 66
siple ado saple of size is easoable. Fo oe details about RSS, see Kau, et al (199. We assue that akig o the auxiliay vaiate, X, is pefect. The associated vaiate, Y, is the with a eo uless the elatio betwee X ad Y is pefect. Let us deote (X j(i, Y j[i] as the pai of the i th ode statistics of X ad the associated eleet Y i the j th cycle. The the aked set saple is (X 1(1, Y 1[1], (X 1(, Y 1[], (X (1, Y [1],, (X (, Y [],, (X (1, Y [1],, (X (, Y [], To obtai Biases ad Mea squaed eo, we coside T y(i = (μ y(i μ y, T = (μ μ x, T xy(i = (μ μ x (μ y(i μ y, σ y(i = E(Y (i μ i, σ = E(X (i μ i, (1 σ xy = E(Y (i μ y (X (i μ x, } ad T = 0, T y(i = 0,, σ = σ x T, σ y(i = σ y T y(i ( σ xy(i = σ xy T xy(i. } The saple ea of each vaiate based o RSS data ad usig the esults obtaied i Dell ad Clutte (197 ca be defied as follows: with vaiace ad co-vaiace X ( = 1 X j=1 (, } ( j=1 Y [] = 1 Y [] Va(X = σ x Va(Y = σ y 1 T, ( 1 T y[i] } Cov(X, Y = σ xy 1 T xy[i] ( Note that μ ad μ y(i deped o ode statistics fo soe specific distibutios ad these values ca be foud i Aold et al (199.. SOME EXISTING ESTIMATORS FOR THE POPULATION MEAN Fo estiatig the populatio ea Y, the usual atio ad poduct estiatos fo Y, espectively as Y R = y X x, (6 ad thei MSEs upto fist degee of appoxiatio ae MSE(Y R = Y Y P = y, (7 x X [(C x + C y ρc x C y ], (8 MSE(Y P = Y [(C x + C y + ρc x C y ], (9 Saawi ad Muttlak (1996 appoached atio ad poduct estiatos ude RSS as Y R = y [] X x (, (10 Y P = y [] x ( X, (11 ad dived thei MSEs to the fist degee appoxiatio as 67
Y MSE(Y R = [(C x + C y ρc x C y 1 ( Y MSE(Y P = [(C x + C y + ρc x C y 1 ( T μ + T y[i] x T μ + T y[i] x μ T xy[i] ] (1 y μ x μ y μ + T xy[i] ] (1 y μ x μ y Fo estiatig the populatio ea Y, Bahl ad Tuteja (1991 give the atio ad poduct type expoetial estiatos as Y Re = y exp [ X x ], (1 X +x Y Pe = y exp [ x X (1 x +X ], ad deived thei MSEs to the fist degee appoxiatio as MSE(Y Re = Y [(C x + C y ρc x C y ], (16 MSE(Y Pe = Y [(C x + C y + ρc x C y ], (17. PROPOSED ESTIMATORS FOR POPULATION MEAN We defie the followig atio ad poduct type expoetial estiatos fo Y ude RSS, espectively as Y Re = y [] exp [ X x ( ], (18 X +x ( Y Pe = y [] exp [ x ( X (19 x ( +X ], hee we have aked auxiliay vaiate ad thus, thee is a iduced ak i study vaiate. the iduced ak o the study vaiate will be pefect if the coelatio betwee the vaiate is pefect othewise it will be wost if thee will be o coelatio(the wost case will ot affect ou poble sice it has aleady be pove by Dell ad Clutte (197. Theefoe, the MSE of Y Re ad Y Pe usig bi-vaiate Taylo seies expasio give as MSE(Y Re = Y [(C x + C y ρc x C y 1 ( MSE(Y Pe = Y [(C x + C y + ρc x C y 1 ( T X T X + + T y[i] Y T y[i] Y T xy[i] ] (0 X Y + T xy[i] ] (1 X Y Pepositio: Let W = μ μ i ad W μ y[i] = μ y[] μ i ad also usig the esult fo Dell ad Clutte (197 i μ i the above equatio ay be witte as MSE(Y Re = Y [(C x + C y ρc x C y 1 ( W + W y[i] W W y[i] ] Y = [(C x + C y ρc x C y 1 (W W y[i] ] = MSE(Y Re Y (W W y[i] It is clea that ( W W y[i] is geate tha zeo. Hece MSE(Y Re MSE(Y Re. ( Also, it ca be poved i siila ways that MSE(Y Pe MSE(Y Pe. (.1 Geealized Expoetial Estiatos Usig RSS We popose a atio-cu-poduct type expoetial estiatos usig RSS as 68
( X Y G = y [] exp [ α x ( ( X α x ( 1 +1 ], ( whee α is soe suitable eal ube whose values ake the iiu MSE of Y G. It ca also be oticed that fo α = 1 ad α = 1 the above equatio becoes Bahl ad Tuteja (1991 usual atio ad poduct expoetial estiatos espectively. Agai usig Taylo seies expasio we get the MSE of Y G as MSE(Y G = Y C x [(α ραc x C y + C y 1 ( α T X 69 αt xy[i] X Y + T y[i] ] ( Y I ode to get the iiu MSE we diffeetiate the above equatio ( by α ad equate it with 0. Hece we get optiu value of α as α opt = ( ρc xc y T xy[i] X Y C x T X Usig the above esult we get the iiu MSE of Y G as. A SIMULATION STUDY MSE(Y G i = Y [C Y T y(i Y. (6 (ρc Y C X T xy[i] X Y (C T ]. (7 X X To illustate how oe ca gai the isight i the applicatio o the popeties of the poposed estiato a copute siulatio was coducted. Bivaiate ado obsevatios wee geeated fo a bivaiate oal distibutio with paaetes μ y, μ x, σ x, σ y ad coelatio coefficiet ρ. The saplig ethod explaied above is used to pick a RSS data with sets of size ad afte epeated cycles to get a RSS of size. A saple of size bivaiate uits is adoly chose fo the populatio (we efe to these data as SRS data. The siulatio was pefoed with =,, ad with = ad 6 (i.e., with total saple sizes of 9, 1, 1, 18, ad 0 fo the RSS ad SRS data sets. Hee, we have aked the auxiliay vaiate X which iduces akig i study vaiate Y (akig o Y will be pefect if ρ = 1 o will be with eos i akig if ρ < 1. Usig R softwae we have coducted,000 eplicatios fo estiates of the eas ad ea squae eos. Results of these siulatios ae suaized by the pecetage elative efficiecies of the estiatos usig the foula. PRE [, Y R ] = MSE(Y R 100 (8 whee, = Y Re, Y Pe, Y G. MSE( PRE [, Y P ] = MSE(Y P 100 (9 MSE( Table 1: Pecetage Relative Efficiecies (PREs of diffeet estiatos of Y with espect to Y R. ρ = 0. ρ = 0.6 ρ = 0.7 Y R Y Re Y G Y R Y Re Y G Y R Y Re Y G 100.00 180.96 78.69 100.00 176.60 19.60 100.00 86.8 1.17 6 100.00 19.98 197.16 100.00 69. 17.10 100.00 00.7 60.1 9.0 100.00 88.08 10.7 100.00 19.1 100.00 1.18 7.96 6 100.00 17.81 18. 100.00 79.89 177.0 100.00.8 199.0 01.1 100.00 7.19 8.1 100.00 71.1 100.00 9.60 09.80 6. 6 100.00 9.0 100.00 8.1 611.9 100.00 80.1 1.18 0 Y R ρ = 0.8 ρ = 0.9 ρ = 0.99 Y Re Y G Y R Y Re Y G Y R Y Re Y G
100.00 111.1 87. 100.00 60.10 19.8 100.00 16.90 10.8 6 100.00 6.7 10.16 100.00 6. 76.76 100.00 7.1 16.7 6611.8 100.00 1.8 716.66 100.00.09 100.00 106.97 878.19 9 6 100.00 7.8 17.67 100.00 8.06 0.6 100.00 11.98 68. 100.00. 887.9 100.00 16.6 771.6 100.00 1.9 1.16 6 100.00 7.0 66.7 100.00 60.0 106. 100.00 10.8 181.6 Table : Pecetage Relative Efficiecies (PREs of diffeet estiatos of Y with espect to Y P. ρ = 0. ρ = 0.6 ρ = 0.7 Y P Y Pe Y G Y P Y Pe Y G Y P Y Pe Y G 100.00 100.70 11.8 100.00 16.76 617.1 100.00 198.68 18.1 6 100.00.90 91.79 100.00 1. 1. 100.00 7.09 1.68 100.00 96.6 101. 100.00 68.88 100.66 100.00 9.6.1 6 100.00 67.0 996. 100.00 18.7.87 100.00 0.86 676.77 100.00 6.8 79.76 100.00 8.69 100.1 100.00 61.9 1898. 6 100.00 16. 0. 100.00 8.1 100.08 100.00 10..7 Y P ρ = 0.8 ρ = 0.9 ρ = 0.99 Y Pe Y G Y P Y Pe Y G Y P 100.00.97 67.77 100.00 18.8 61.0 100.00 8.7 6979.8 6 100.00 10.8 6.8 100.00 1. 77.1 100.00 199. 980.6 100.00 9.0 160.8 100.00 07.98 998.6 100.00 17.07 979. 6 100.00 11.7 669.86 100.00 111.68 10.7 100.00 11.9 1768.70 100.00 0.81 16.19 100.00 1.8 901.0 100.00 17.6 8.1 6 100.00 16.10 998.60 100.00 1.16 6096.8 100.00 176.6 1788.9 Y Pe Y G. CONCLUSIONS It is obseved fo Table 1 that the PREs of the poposed atio type expoetial estiato usig ak set saplig, Y Re ad the poposed geealized expoetial estiatos usig ak set saplig Y G ae oe efficiet copaed to the existig Saawi ad Mutlak(1996 atio estiatoy R. Also fo Table, it ca be obseved the PREs of the poposed poduct type expoetial estiato usig aked set saplig, Y Pe ad the poposed geealized expoetial estiatos usig aked set saplig Y G ae oe efficiet copaed to the existig poduct estiato Y Pe. Fially fo Tables 1 ad we ca coclude that the poposed estiatos Y Re, Y Pe ad Y G ae oe appopiate estiatos tha the existig popula estiatos Y R, Y P, Y R ad Y P has appeciable efficiecy. Ackowledgeet: The authos ae thakful to the edito ad the leaed efeees fo thei valuable coets ad suggestios towads the ipoveet of the pape. RECEIVED JUNE, 016 REVISED SEPTEMBER, 016 70
REFERENCES [1] AL OMARI, A.I. ad BOUZA, C.N. (01: Ratio estiatos of the populatio ea with issig values usig aked set saplig. Evioetics, 6., 67-76. [] AL-SALEH, M. F., ad AL-OMARI, A. I. (00: Multistage aked set saplig. Joual of Statistical Plaig ad Ifeece, 10 (, 7-86. [] ARNOLD, B.C., BALAKRISHNAN, N. ad NAGARAJA, H.N. (199: A fist couse i ode statistics. SIAM, Vol.. [] BAHL, S. ad TUTEJA, R.K. (1991: Ratio ad poduct type expoetial estiato. Ifoatio ad Optiizatio Scieces, 1, 19-16. [] DELL, T.R. ad CLUTTER, J.L. (197: Raked set saplig theoy with ode statistics backgoud. Bioetics, 8, -. [6] KAUR, A., PATIL, G.P., SINHA, A.K. ad TAILLIE, C. (199: Raked set saplig: a aotated bibliogaphy. Evioetal ad Ecological Statistics,, -. [7] MANDOWARA, V.L. ad MEHTA, N. (01: Efficiet geealized atio-poduct type estiatos fo fiite populatio ea with aked set saplig. Austia Joual of Statistics,, 17-18. [8] McINTYRE, G. A. (19: A ethod fo ubiased selective saplig, usig aked sets. Cop ad Pastue Sciece,, 8-90. [9] SAMAWI, H.M. ad MUTTLAK, H.A. (1996: Estiatio of atio usig ak set saplig. Bioetical Joual,8, 7-76. [10] STOKES, L. (1977: Raked set saplig with cocoitat vaiables. Couicatios i Statistics-Theoy ad Methods,6, 107-111. [11] STOKES, L. (1980: Estiatio of vaiace usig judget odeed aked set saples. Bioetics, 6, -. [1] TAKAHASI, K. ad WAKIMOTO, K. (1968: O ubiased estiates of the populatio ea based o the saple statified by eas of odeig. Aals of the Istitute of Statistical Matheatics, 0, 1-1. [1] WOLFE, DOUGLAS A. (00: Raked set saplig: a appoach to oe efficiet data collectio. Statistical Sciece, 19, 66-6. 71