QUANTILE ESTIMATION IN SUCCESSIVE SAMPLING

Jornal of the Korean Statistical Society 2007, 36: 4, pp 543 556 QUANTILE ESTIMATION IN SUCCESSIVE SAMPLING Hosila P. Singh 1, Ritesh Tailor 2, Sarjinder Singh 3 and Jong-Min Kim 4 Abstract In sccessive sampling on two occasions the problem of estimating a finite poplation qantile has been considered. The theory developed aims at providing the optimm estimates by combining i three doble sampling estimators viz. ratio-type, prodct-type and regression-type, from the matched portion of the sample and ii a simple qantile based on a random sample from the nmatched portion of the sample on the second occasion. The approximate variance formlae of the sggested estimators have been obtained. Optimal matching fraction is discssed. A simlation stdy is carried ot in order to compare the three estimators and direct estimator. It is fond that the performance of the regression-type estimator is the best among all the estimators discssed here. AMS 2000 sbject classifications. Primary 62D05; Secondary 62G05. Keywords. Axiliary information, finite poplation qantile, partial replacement, sccessive sampling. 1. Introdction The problem of qantile estimation often arises when variables with a highly skewed distribtion, sch as income, are stdied. When there is an extensive literatre on the estimation of mean and total in sample srveys, relatively less efforts have been made in the development of efficient procedres for estimating Received May 2006; accepted October 2007. 1 School of Stdies in Statistics, Vikram University, Ujjain 456010, India e-mail: hsingh@winedt.com 2 School of Stdies in Statistics, Vikram University, Ujjain 456010, India 3 Department of Statistics, St. Clod State University, St. Clod, MN, 56301, USA e-mail: sarjinder@yahoo.com 4 Corresponding athor. Statistics Discipline, Division of Science and Mathematics, University of Minnesota at Morris, Morris, MN, 56267, USA e-mail: jongmink@morris.mn.ed

544 Hosila P. Singh et al. finite poplation qantiles. It is well known that the se of axiliary information at the estimation stage can typically increase the precision of estimates of a parameter. A large nmber of estimators for estimating poplation mean based on axiliary information are available in the literatre with their properties nder simple random sampling design and other sampling designs. However, few athors inclding Chambers and Dstan 1986, Kk and Mak 1989, Rao et al. 1990, Mak and Kk 1993, Kk 1993, Reda et al. 1998, Reda and Arcos 2001, Allen et al. 2002, Singh et al. 2001, Singh and Joarder 2002, Singh et al. 2003 and Singh 2003 have discssed the problem of estimating finite poplation qantile sing axiliary information in sample srveys. The problem of sampling on two sccessive occasions was first introdced by Jessen 1942 and related review is available in Biradar and Singh 2001. It is to be mentioned that the stdy relating to environmental isses freqently involves variables with extreme vales which inflence the vale of mean. In sch sitations the estimation of second qantile assmes importance, as it is not affected by the extreme vales. This led athors to consider the problem of estimation of qantile nder sccessive sampling. In this paper, we have sggested three estimators viz. i ratio-type ii prodct-type and iii regression type with their properties. 2. Sggested Estimators in Sccessive Sampling Consider a finite poplation Ω = {U 1, U 2,..., U N } of N identifiable nits which is spposed to be sampled on two occasions. Assme that size of the poplation remains nchanged bt vales of nit change over two occasions. Let y i x i be the vale of the variate nder stdy for the i th nit of the second first occasion. On the first occasion an initial sample of size n 1 is selected by simple random sampling withot replacement SRSWOR scheme. Ot of these n 1 nits, m nits called a matched sample are retained on the second occasion while a fresh simple random sample of size = n m called a nmatched sample is drawn withot replacement on the second occasion from the remaining N n 1 nits of the poplation so that the total sample size at the second occasion becomes n = m +. Let y 1, y 2,..., y N be the vales of the poplation elements U 1, U 2,..., U N, for the variable of interest y. For any y < y <, the poplation distribtion fnction F y y is defined as the proportion of elements in the poplation that are less than or eqal to y. The finite poplation β-qantile

Qantile Estimation in Sccessive Sampling 545 of y is given by Q y β = inf{y; F y y β} = Fy 1 β. The problem nder investigation is to estimate the poplation qantiles Q y β of order β 0 < β < 1 on the crrent second occasion. Let ˆQ x β = inf{x; ˆF x x β} be the sample qantile of order β on the first occasion and ˆQ y β = inf{y; ˆF y y β} the sample qantile on the crrent second occasion, noting that ˆF x x ˆF y y is a monotone nondecreasing fnction xy. Denote by ˆQ xm β and ˆQ ym β be the sample qantiles of the matched sample on the first and second occasions respectively, and ˆQ y β the sample qantile of the nmatched sample on the crrent occasion. For estimating the poplation qantile Q y β based on sccessive sampling, two independent estimators can be made. First, based on sample of size drawn fresh on the crrent occasion and second based on the sample of size m common to both the occasions. Ths we define the following estimators: i ˆQ r = α ˆQ r ym β + 1 α ˆQ y β, ratio-type ii ˆQ p = δ ˆQ p ym β + 1 δ ˆQ y β, prodct-type iii ˆQ l = γ ˆQ l ym β + 1 γ ˆQ y β, regression-type where α, δ and γ are sitably chosen scalars, { } ˆQ r ym = ˆQ ˆQx β ym β, ratio-type ˆQ xm β { } ˆQxm ˆQ p ym = ˆQ β ym β, prodct-type ˆQ x β ˆQ l ym = ˆQ ym β + b{ ˆQx β ˆQ } xm β, regression-type 2.1 with b = ˆf x ˆQxm β ˆf y ˆQym β { } ˆPxym β1 β 1, where ˆP xym is the proportion of elements in sample sch that y ˆQ ym β and x ˆQ xm β, ˆfx ˆQxm β and ˆf y ˆQym β are the estimates of f x Qx β

546 Hosila P. Singh et al. and f y Qy β respectively determined by the method as adopted by Silverman 1986 where f y f x is the derivative of F y F x, the limiting vale of F y F x as N, for instance, see Randles 1982 and Rao et al. 1990. Ths we note that as N the distribtion of the bivariate variable x, y approaches a continos distribtion with marginal densities f y and f x for y and x respectively, see Kk and Mak 1989, p. 264. The variances of ˆQ r, ˆQp and ˆQ l are respectively given by V ˆQr = V ˆQy β 2αV ˆQy β ], + α 2{ } V ˆQy β + V ˆQr ym V ˆQp = V ˆQy β + α 2{ } V ˆQy β + V ˆQp ym ] +2αV ˆQy β and V ˆQl = V ˆQy β + α 2{ } V ˆQy β + V ˆQl ym +2αV ˆQy β ], where the variances of ˆQy β, ˆQr l ym β, ˆQp ym β and ˆQ ym β to the first degree of approximation or alternatively, of order n 1 are respectively given by V ˆQy β = 1 f β1 β 2 = 1 f Ay, β, 2.2 f y Q y β V ˆQr ym β = Ay, β V ˆQp ym β = Ay, β V ˆQl ym β = Ay, β 1 f m m + 1 f m m + 1 f m m ] 1 m 1 θθ 2ρ c, n 1 ] 1 m 1 θθ + 2ρ c, n 1 ] 1 m 1 ρ 2 c, n 1 where f = /N, f m = m/n, Ay, β = {β1 β}/{f y Q y β 2 }, θ = {Q y β f y Q y β}/{q x βf x Q x β}, ρ c = {P xy /β1 β 1} and P xy denotes the

Qantile Estimation in Sccessive Sampling 547 proportion of nits in the poplation with x Q x β and y Q y β. See, Kk and Mak 1989, p. 268. Ths we get the variances of ˆQ r, ˆQ p and ˆQ l to the first degree of approximation as 1 f 1 2α V ˆQr = Ay, β V ˆQp = Ay, β V ˆQl = Ay, β which are respectively minimized for { + α 2 1 f + 1 f m m }] 1 + m 1 θθ 2ρ c, 2.3 n 1 { 1 f 1 2δ + δ 2 1 f + 1 f m m }] 1 + m 1 θθ + 2ρ c, 2.4 n 1 { 1 f 1 2δ + γ 2 1 f }] + 1 f m m 1 m 1 ρ 2 c, 2.5 n 1 α = δ = γ = 1 f / ], 2.6 1 f + 1 f m m + 1 θθ 2ρ c 1 f / ], 1 f + 1 f m m + 1 θθ + 2ρ c 1 f / ]. 2.7 1 f + 1 fm m 1 ρ 2 c Hence by inserting 2.6 to 2.7 in 2.3 to 2.5 the reslting minimm

548 Hosila P. Singh et al. variances of ˆQ r, ˆQp and ˆQ l are respectively gived by ] 1 f m min V ˆQ r = 1 f m + 1 θθ 2ρ c Ay, β ], 2.8 1 f + 1 f m m + 1 θθ 2ρ c ] 1 f m min V ˆQ p = 1 f m + 1 θθ + 2ρ c Ay, β ], 2.9 1 f + 1 fm m + 1 θθ + 2ρ c ] 1 f m min V ˆQ l = 1 f m 1 ρ 2 c Ay, β ]. 2.10 1 f + 1 f m m 1 ρ 2 c In the next section, we will consider analytical comparisons of the proposed estimators. 3. Analytical Comparisons From 2.2, 2.8, 2.9 and 2.10 we have V ˆQy β min V ˆQ r = 1 f 2 2 Ay, β D r 0, 3.1 V ˆQy β min V ˆQ p = 1 f 2 2 Ay, β D p 0, 3.2 V ˆQy β min V ˆQ l = 1 f 2 2 Ay, β D l 0, 3.3 min V ˆQyr β min V ˆQ l = 1 f 1 m 1 Ay, βθ ρc 2 0, 3.4 n 1 D l D r min V ˆQyp β min V ˆQ l = 1 f 1 m 1 Ay, βθ + ρc 2 0, 3.5 n 1 D l D r

Qantile Estimation in Sccessive Sampling 549 where D r = D p = D l = 1 f + 1 f m 1 m + 1 f + 1 f m 1 m + 1 f + 1 f m 1 m ] θθ 2ρ c, ] θθ + 2ρ c, ] ρ 2 c. Expressions 3.1, 3.2 and 3.3 clearly indicate the redction in variance de to se of ˆQr, ˆQp and ˆQ l respectively instead of ˆQy β as an estimator of Q y β. Frther the redction in variance by sing the estimator ˆQ l instead of ratio-type estimator ˆQ r and the prodct-type estimator ˆQ p as estimators of Q y β are given by 3.4 and 3.5 respectively. From 3.1 to 3.2 we have the following ineqalities, min V ˆQ l min V ˆQ r V ˆQ y β, 3.6 min V ˆQ l min V ˆQ p V ˆQ y β. 3.7 It follows from 3.6 and 3.7 that the regression estimator ˆQ l is better than ˆQ y β, ˆQr and ˆQ p. In the case n 1 = n i.e. sample sizes are the same at both the occasions and assme that the poplation size N is large enogh so that f 0, f m 0, the expressions in 2.2, 2.3, 2.4 and 2.5 respectively redce to V ˆQy β Ay, β =, V Ay, β ˆQr = mn1 2α + α 2{ n 2 + 2 θθ 2ρ c } ], 3.8 mn V Ay, β ˆQp = mn1 2δ + δ 2{ n 2 + 2 θθ + 2ρ c } ], mn V ˆQl = Ay, β mn mn1 2γ + γ 2{ n 2 2 ρ 2 c} ]. 3.9 Ths the variance expressions 3.8 to 3.9 are respectively minimized with

550 Hosila P. Singh et al. 2.6 to 2.7 for α = δ = γ = mn n 2 + 2 θθ 2ρ c, mn n 2 + 2 θθ + 2ρ c, mn n 2 2 ρ 2. c Ths the reslting variances of ˆQ r, ˆQp and ˆQ l are respectively given by ] n + θθ 2ρ c min V ˆQ r = Ay, β ], 3.10 n 2 + 2 θθ 2ρ c ] n + θθ + 2ρ c min V ˆQ p = Ay, β ], 3.11 n 2 + 2 θθ + 2ρ c ] min V ˆQ n ρ 2 c l = Ay, β ]. 3.12 n 2 2 ρ 2 c Minimization of 3.10, 3.11 and 3.12 with respect to gives the optimm vale of respectively as n = 1 + 1 + θθ 2ρ c, n = 1 + 1 + θθ + 2ρ c, n = 1 +. 3.13 1 ρ 2 c Ths the reslting vales of min V ˆQ r, min V ˆQ p and min V ˆQ l are respectively given by min V ˆQ r min V ˆQ p min V ˆQ l opt opt opt Ay, β = 1 + ] 1 + θθ 2ρ c, 3.14 2n = Ay, β 2n = Ay, β 2n 1 + ] 1 + θθ + 2ρ c, 3.15 1 + ] 1 ρ 2 c. 3.16

Qantile Estimation in Sccessive Sampling 551 Frther the variance of the direct estimator ˆQ yn β to the first degree of approximation is given by V ˆQβ yn = 1 f n Ay, β, 3.17 n when the poplation size N is very large so that f n 0, we get V ˆQβ yn = Ay, β. 3.18 n From 3.14, 3.15, 3.16, and 3.18 we have V ˆQyn β min V ˆQ r opt Ay, β θθ 2ρ c = { 2n 1 + } > 0, if ρ c > θ 1 + θθ 2ρ c 2, 3.19 V ˆQyn β min V ˆQ p opt Ay, β θθ + 2ρ c = { 2n 1 + } > 0, if ρ c < θ 1 + θθ + 2ρ c 2, min V ˆQ r opt min V ˆQ l opt Ay, β θ ρ c = { 2 1 2n + θθ 2ρc + 0, 1 ρc} 2 min V ˆQ p opt min V ˆQ l opt Ay, β θ + ρ c = { 2 1 2n + θθ + 2ρc + 0. 3.20 1 ρc} 2 From 3.19 to 3.20 we have the following ineqalities min V ˆQl opt min V ˆQr opt V ˆQyn, if ρc > θ 2, min V ˆQl opt min V ˆQp opt V ˆQyn, if ρc < θ 2. Finally we conclde that the regression-type estimator ˆQ l has the least variance and hence more efficient than ˆQ r, ˆQp and ˆQ yn. For = 0 complete

552 Hosila P. Singh et al. matching or = n no matching the variance in 3.10, 3.11 and 3.12 redces to: min V ˆQr = min V ˆQp = min V ˆQl = V ˆQyn β = Ay, β. n Ths in this case all the estimators ˆQ yn β, ˆQr, ˆQp and ˆQ l are eqally efficient. Frther from 3.16 and 3.17, the efficiency of ˆQl with respect to direct estimator ˆQ yn is given by We also note from 3.13 that Ths we have E = 2 1 + 1. 1 ρc 2 n = 1 λ = µ = 1 + 1. 1 ρc 2 3.21 m n = λ = 1 ρ 2 { c 1 + }. 1 ρ 2 c The eqation 3.21 shows that the optimm percentage to be matched decreases with increasing vale of ρ c. For ρ c = 1, which implies P xy = 2β1 β, this percentage is 0 and 50 respectively. However, for m = 0, b in 2.1 cannot be obtained and the reslts derived above are invalid. It is expected that ρ c lies between 0.50 and 1. The percentage gain in efficiency E 1100% increases with increasing vale of ρ c for optimm matching percentage. Remark 3.1. For β = 0.5 the stdies of this paper redce to the estimation of poplation median in sccessive sampling. 4. Simlation Stdy: Three Qartiles of Abortion Cases For the prpose of simlation stdy we consider the sitation of a poplation consisting of N = 50 states, and let y i represent the nmber of abortions dring 2000 and x i be the nmber of abortions dring 1992 in the i th state. Table 4.1 gives the descriptive statistics of nmber of abortions dring 1992 and 2000. The vale of the correlation between the nmber of abortions dring 1992 and 2000 is fond to be ρ xy = 0.987. The following graphs in Figre 4.1 and Figre 4.2 show that the distribtion of the nmber of abortions in different states is skewed

Qantile Estimation in Sccessive Sampling 553 Table 4.1 Descriptive statistics of nmber of abortions dring 1992 and 2000 Abortions 1992 Abortions 2000 Mean 30.6 26.3 Standard Error 7.3 6.0 Median 14.5 12 Mode 7 6 Standard Deviation 51.3 42.8 Krtosis 18.0 13.1 Skewness 3.9 3.4 Minimm 1 1 Maximm 304 236 Cont 50 50 towards right. One reason of skewness may be the distribtion of poplation in different states, that is, the states having larger poplations are expected to have larger nmber of abortion cases. Ths skewness of the data indicates that the se of three qartiles may be a good measre of central locations than mean in sch a sitation. We selected 5,000 samples of n 1 = 20 sing withot replacement sampling and only the nmber of abortions dring 1992 among the selected states was noted. Ths ˆQ xnl l k, l = 1, 2, 3 and k = 1, 2,..., 5000 sample qantiles were compted. From each one of the selected 5,000 samples, we decided to retain m = 5 states in each sample, and we selected new = n m = 10 5 = 5 states ot of N n 1 = 50 20 = 30 states sing withot replacement sampling. Figre 4.1 Abortions dring 1992 verss nmber of states.

554 Hosila P. Singh et al. Figre 4.2 Abortions dring 2000 verss nmber of states. From the m nits retained in the sample, we compted ˆQ xm l k, ˆQ ym l k with l = 1, 2, 3 for k = 1, 2,..., 5000; and from the new nmatched nits selected on the second occasion we also compted ˆQ y l k with l = 1, 2, 3 for k = 1, 2,..., 5000. We decided to select a parameter Φ between 0.1 and 0.9 with a step of 0.1. Then the relative efficiencies of the ratio type estimators, for l = 1, 2, 3, { } ˆQxn ˆQ R l k = Φ ˆQ l k ym l k + 1 Φ ˆQ ˆQ y l k, 4.1 xm l k with respect to ˆQ y l k are given by REl = ] 2 5000 k=1 ˆQ y l k Q y l 5000 k=1 ] 2 100 for l = 1, 2, 3, 4.2 ˆQ R l l Q y l where Q y l for l = 1, 2, 3 denotes the l th poplation qartile. In Table 4.2, the relative efficiency of the ratio type estimator of the first qartile Q y 1 ranges from 111% to 160% with median efficiency being 146%; the relative efficiency of the estimator of second qartile Q y 2 ranges from 110% to 159% with median efficiency being 148%; and the relative efficiency of the estimator of third qartile Q y 3 ranges from 114% to 281% with median efficiency of 197%. It is interesting to note that the relative efficiency of the first and second

Qantile Estimation in Sccessive Sampling 555 Table 4.2 Relative efficiency of the ratio type estimators Φ RE1 RE2 RE3 0.1 111.90 110.39 114.23 0.2 124.63 121.02 131.34 0.3 135.82 131.68 150.78 0.4 146.90 141.36 172.22 0.5 155.15 148.57 197.33 0.6 159.32 156.02 223.28 0.7 160.05 157.96 249.17 0.8 158.30 159.49 272.42 0.9 146.72 154.92 281.85 qartiles behaves in the same fashion that as the vale of Φ increases from 0.1 to 0.9 the relative efficiency increases at first and then starts decreasing, whereas the relative efficiency of the estimator of the third qartile goes on increasing. It is not obvios to find its reason, bt one reason may be that data is skewed to right, and the ratio type adjstment may be making more sense than the simple sample qartile estimator. We acknowledge that more simlation may be performed in ftre stdies as pointed ot by one of the learned referees. 5. Conclsion To or knowledge, this is a first attempt to estimate finite poplation qartiles sing sccessive sampling. The analytical and empirical reslts spport the fact that estimation of three qartiles sing sccessive sampling is feasible, which was ignored by the srvey statisticians in the past. Acknowledgements The athors are thankfl to the editor, associate editor and the two referees for the valable comments on the original version of this manscript which lead to sbstantial improvement. The athors also acknowledge the se of free access to the data by the Statistical Abstracts of the United States 2006 sed here to evalate one methodology over the other. References Allen, J., Singh, H. P., Singh, S. and Smarandache, F. 2002. A general class of estimators of poplation median sing two axiliary variables in doble sampling, In

556 Hosila P. Singh et al. Randomness and Optimal Estimation in Data Sampling Khoshnevisan, M. et al., eds., 26 43, American Research Press, Rehoboth. Biradar, R. S. and Singh, H. P. 2001. Sccessive sampling sing axiliary information on both the occasions, Calctta Statistical Association Blletin, 51, 243 251. Chambers, R. L. and Dnstan, R. 1986. Estimating distribtion fnctions from srvey data, Biometrika, 73, 597 604. Jessen, R. J. 1942. Statistical investigation of a sample srvey for obtaining farm facts, Iowa Agricltral Experiment Statistical Research Blletin, 304, 1 104. Kk, A. Y. C. 1993. A kernel method for estimating finite poplation distribtion fnctions sing axiliary information, Biometrika, 80, 385 392. Kk, A. Y. C. and Mak, T. K. 1989. Median estimation in the presence of axiliary information, Jornal of the Royal Statistical Society, Ser. B, 51, 261 269. Mak, T. K. and Kk, A. Y. C. 1993. A new method for estimating finite poplation qantiles sing axiliary information, The Canadian Jornal of Statistics, 21, 29 38. Randles, R. H. 1982. On the asymptotic normality of Statistics with estimated parameters, The Annals of Statistics, 10, 462 474. Rao, J. N. K., Kovar, J. G. and Mantel, H. J. 1990. On estimating distribtion fnctions and qantiles from srvey data sing axiliary information, Biometrika, 77, 365 375. Reda Garcia, M., Arcos Cebrian, A. and Artes Rodrigez, E. 1998. Qantile interval estimation in finite poplation sing a mltivariate ratio estimator, Metrika, 47, 203 213. Reda Garcia M. and Arcos Cebrian, A. 2001. On estimating the median from srvey data sing mltiple axiliary information, Metrika, 54, 59 76. Silverman, B. W. 1986. Density Estimation for Statistics and Data Analysis, Chapman and Hall, London. Singh, H. P., Singh, S. and Pertas, S. M. 2003. Ratio type estimators for the median of finite poplations, Allgemeines Statistisches Archiv, 87, 369 382. Singh, S., Joarder, A. H., and Tracy, D. S. 2001. Median estimation sing doble sampling, Astralian and New Zealand J. Statist., 43, 33 46. Singh, S. and Joarder, A. H. 2002. Estimation of the distribtion fnction and median in two phase sampling, Pakistan Jornal of Statistics, 18, 301 319. Singh, S. 2003. Advanced Sampling Theory with Applications: How Michael Selected Amy, Klwer Academic Pblishers, The Netherlands.