Estimation of the Population Variance Using. Ranked Set Sampling with Auxiliary Variable

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1 Int. J. Contep. Math. Sciences, Vol. 5, 00, no. 5, Estiation of the Population Variance Using Ranked Set Sapling with Auiliar Variable Said Ali Al-Hadhrai College of Applied Sciences, Nizwa, Oan Abstract When the variable of interest Y is correlated to another variable X that has soe inforation, then the use of auiliar variable a iprove the efficienc of the estiation of the variable under the stud. In this paper, ratio estiator for the population variance based on Ranked Set Sapling ( was considered. The suggested estiator was copared to the corresponding one based on Siple Ro Sapling (SRS. Coputer Siulation fro soe distributions showed that estiator has saller bias Mean Square Error (MSE than that fro SRS. Kewords; Ranked Set Sapling; Ratio Estiator; Population variance; efficienc Introduction Ranked Set Sapling ( was first used b McIntre (95 to estiate the population ean. The ethod is ore efficient when estiating the population ean. However, the gain of over SRS is little when estiating the population variance. Stokes (980 studied estiation of the population variance using a bias estiator of was defined but it is asptoticall unbiased. Yu et al. (999 studied soe unbiased estiates of in the paraetric case of a noral population. Tiwari Kva (00 proposed unbiased estiator for for location-scale failies of setric distribution. MacEachern (00 developed an unbiased estiator of the variance of a population based on. The suggested estiator is better than estiating the variance based on SRS ore efficient than the estiator based on proposed b Stokes (980. Tiensuwan Sarikavanij (00 proposed two

2 568 S. A. Al-Hadhrai unbiased estiators of the variance for ultiple ccles under balanced case. Their proposed estiators are ore efficient than the one based on SRS. The showed that there is no unbiased estiate of the population variance based on a single ccle of. Perron Sinha (004 showed that for ore than one ccle, it is possible to construct a class of quadratic unbiased estiates of in both balanced unbalanced cases. The derived a iniu variance unbiased quadratic nonnegative estiate of within a certain class of quadratic estiates. Ahad (004 suggested soe bootstrap techniques for estiation of variance under. Sengupta Mukhuti (006 proposed soe unbiased estiators of the variance of the eponential distribution. In this paper ratio estiator for the population variance is considered copared to the ratio estiator based on SRS. Ratio Estiator For the Population Variance in SRS Izaki(98 suggested the following estiator for the population variance where s respectivel, defined as s s ( s / s S ( s are unbiased estiators of population variance n i ` ( i n 4 K μ04 / μ0 s n i ` ( i n S The MSE of the estiators is MSE ( s λs K + K θ where λ /n, K μ40 / μ0,, θ μ / μ0μ0, N r s rs (/ N ( ( i i i μ μ μ S, Soe Estiators for the Population Variance Based on Stokes (980 suggested for one ccle the following estiator for where X (/ (, i i ˆ i ( ( i

3 Estiation of the population variance 569 with + ( i i E( ˆ (/ ( ( μ μ The variance of the estiator is Var( ˆ μ + 4 τ + 4 τ μ ( i i i 4 ( 4 + ( i ( i ( i ( i< i i Stokes(980 suggested for r ccles the following estiator for with ˆ 4( i ( i ( i ( i ( i r j i ( ( j ( i r + r ( i i E( ˆ (/ ( ( μ μ The variance of the estiator is r r r Var( ˆ μ4( i + 4 τ( i ( i + 4 τ( i μ( i ( r i i r i r 4 ( r ( r 4 + ( i ( i ( i ( r i< i r i The estiators provided b Stokes are biased. Montip & Sukuan(00 showed that for one ccle there is no unbiased estiator for but for ore than one ccle the proposed two unbiased estiators for. The first estiator is W B ˆ + r ( r r ( ( ( i, j i j j where W ( X X r B ( X X j ( j The also showed that the variance of the estiator is X (/ r X r j ( ( j T+ 4T + 4T Var( ˆ T 5 + ( i r r r( r i i i ( i ( i where A μ4( i, A τ, A τ ( i μ( i, μ E( X μ k τ ( i μ( i μ k ( i ( i ( i i A 4 4 i ( i, The second estiator is

4 570 S. A. Al-Hadhrai ˆ * r + B W + r( r r * ( r * ( ( i i j i, j where W ( X X The variance of the estiator is * ( ( i j B r X X A+ 4A + 4A r r Var( ˆ A ( i r r ( r r i The also coputed the variance of the above estiators when the underling distribution is unifor, eponential noral. For ore details refer to Montip & Sukuan (00. 4 Ratio Estiator for the Population Variance Based on In this section we assue that the population variance of X is known, X. So we can define a ratio estiator of the variance of Y as follows ˆ R (4 ˆ where ˆ ˆ are unbiased estiators defined in equation (. General for of Talor series for a bivariate function hxy (, is hxy (, h( μ, μ + h( μ, μ ( Y μ + h( μ, μ ( Xμ + + ( Y μ h( μ, μ ( X μ( Y μ h( μ, μ + ( X μ h( μ, μ where h hxy (, hxy (, ( μ, μ μ, μ h ( μ, μ μ, X Y X μ. So, appling this epansion to ˆR about gives bias zero when first order epansion is used. However, the second order bivariate Talor epansion gives

5 Estiation of the population variance 57 ˆ ˆ ˆ R + ( ( Then the bias of the estiator is ( ( ˆ ( ˆ ( ˆ + Bias ( ˆ E ˆ E ˆ ˆ This can be epressed as R ( ( ( ( Bias ( ˆ Var ˆ Cov ˆ, ˆ ( about ˆ ˆ ( ( Using first order of the Talor epansion of, then ˆ ( ˆ ( ˆ + Then the variance of the estiator is Var( ˆ ( ˆ ( ˆ ( ˆ, ˆ Var Var Cov ˆ is 5 Siulation Stud The behavior of the proposed estiator has been investigated nuericall. Let us assue that the variable of interest Y a concoitant variable X were correlated with a correlation coefficient ρ. The saples were generated fro three distributions; bivariate noral, bivariate eponential, bivariate gaa. For generating bivariate noral, S-Plus codes were used since it is alread provided but both eponential gaa were generated b using the iture approach ethod proposed b Minhajuddin et al (004. The algorith started with generating fro a negative binoial as a prior distribution. Then, conditional on that value, bivariate saple were generated fro the posterior distribution. For each estiator, 0000 saples generated fro the distributions the estiates of were calculated as follows: For, the estiator ( ˆ / ˆ R was used where

6 57 S. A. Al-Hadhrai W B ˆ + r ( r W B ˆ + r ( r And for SRS, we used the estiator ( ˆ / ˆ. The average of ˆ ' RSRS SRS SRS s the Mean Square Error (MSE were calculated respectivel using Eˆ 0000 ˆ 0000 i i Varˆ ( ˆ i i Different values for ρ were used. Table (,( ( show the bias the MSE of the estiators fro noral distribution, eponential gaa distribution respectivel. Table ( Siulation results fro noral with μ μ. 5, 6, Nuber of ccles with different set size correlation coefficient ρ ρ Bias MSE efficienc SRS SRS

7 Estiation of the population variance 57 Table ( Siulation results fro eponential with ean for both X Y. Nuber of ccles with different set size correlation coefficient ρ ρ Bias MSE SRS SRS efficienc

8 574 S. A. Al-Hadhrai Table ( Siulation results fro gaa (, Nuber of ccles with different set size correlation coefficient ρ. ρ Bias MSE SRS SRS efficienc Concluding Rearks Fro the siulation results given in the tables the following can be concluded: - Both the bias MSE of all the estiators decrease when either the saple size or the correlation coefficient increases.

9 Estiation of the population variance Both the bias MSE of estiators are saller than the corresponding ones based on SRS. The relative efficienc of estiators with respect to the corresponding one based on SRS is greater than. - Both the bias MSE of the estiators for noral distribution variance are saller than those fro eponential gaa. 4- The estiators of the variance of eponential distribution have greater bias MSE than the one for gaa under the given paraeters unless the correlation coefficient is near. References [] T. Ahad, Bootstrap Techniques for estiation of variance under ranked set sapling, Atlas conference, institute of engineering technolog, India (004: 7-9 Deceber. [] A.T.M. Minhajuddin, I.R. Harris W.R. Schucan, Siulating ultivariate distributions with specific correlations, Journal of Statistical Coputation Siulation, 74(4( [] S.N. MacEachern, O. Ozturk, D.A. Wolf, G.V. Shark, A new ranked set saple estiator of variance.journal of the Roal Statistical Societ 64((00, [4] G.A. McIntre, A ethod of unbiased selective sapling, using ranked sets, Australian J. Agricultural Research, (95, [5] F. Perron,,B.K. Sinha, Estiation of Variance based on a ranked set saple, Journal of statistical planning inference, 0(004, -8 [6] S.L. Stokes, Estiation of variance using judgent order ranked set saples, Bioetrics 6(980, 5-4. [7] S. Sengupta, S. Mukhuti, Unbiased Variance estiation in a saple eponential population using ranked set saples. Journal of statistical planning inference. 6(4 (006, [8] R.C. Tiwari, P.H. Kva, Ranked Set Sapling fro location-scale failies of setric distributions, Counications in Statistics: Theot Methods 0(00, [9] M. Tiensuwan, S. Sarikavanij, On estiation of the population variance based on ranked set saple, Journal of Applied Statistical Science, (4(00, 8-95.

10 576 S. A. Al-Hadhrai [0] P.L.H. Yau, K. La, B.K. Sinha, Estiation of noral variance based on balanced unbalanced Ranked Set Saples, Environental Ecological Statistics, 6(999, -45 Received: June, 00

Estimation of the Population Mean Based on Extremes Ranked Set Sampling

Estimation of the Population Mean Based on Extremes Ranked Set Sampling Aerican Journal of Matheatics Statistics 05, 5(: 3-3 DOI: 0.593/j.ajs.05050.05 Estiation of the Population Mean Based on Extrees Ranked Set Sapling B. S. Biradar,*, Santosha C. D. Departent of Studies