characterization of ranked set sampling bayes estimators with ...

SOOCHOW JOURNAL OF MATHEMATICSVolume 28, No. 2, pp. 223-234, April 2002CHARACTERIZATION OF RANKED SET SAMPLINGBAYES ESTIMATORS WITH APPLICATION TO THENORMAL DISTRIBUTIONBYMOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASAbstract. McIntyre’s (1952) concept of ranked set sampling (RSS) was founduseful for increasing the efficiency of estimating the population mean, when the distributionis unknown. The procedure is applicable when the observations are easierto be ranked than measured. In this paper, Bayesian estimation of the parameter ofa distribution using RSS is considered. Using the notion of multiple imputations, acharacterization that relates the Bayes estimators using RSS to those using simplerandom sample (SRS) is obtained. This characterization turns out to be useful forstudying some of the properties of these estimators, which have usually complicatedforms. It is also used to approximate these estimators. The characterizationis applied to the case of normal distribution with conjugate prior.1. IntroductionRanked set sampling (RSS) was first introduced by McIntyre (1952). Theaim was to improve the efficiency of the sample mean as an estimator of thepopulation mean (µ). RSS is largely used in situation where the sample unitscould be easily arranged with respect to the variable of interest or any otherconcomitant variable than quantified. McIntyre’s concept depends on measuring,the first visually ordered unit from the first set, the second visually ordered unitfrom the second set and so on until the maximum unit from the last set, whereeach set is of size m units. Let X (ii) denote the ith-quantified unit from the ithset i = 1, . . . , m, then {X (11) , . . . , X (mm) } is called a ranked set sample.Received February 16, 2001; revised September 14, 2001; revised October 23, 2001.Key words. ranked set sampling, Bayes estimator, Bayes risk, efficiency, multiple imputation,simple random sampling.223The

224 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASwhole procedure can be repeated, if necessary, r times to obtain a RSS of sizerm units. McIntyre proposed the mean of these m units, denoted by ⌢ µ RSS as acompetitor estimator of µ to the mean of a simple random sample (SRS) of sizem, denoted by ⌢ µ SRS .Takahasi and Wakimoto (1968) provided the statistical theory for RSS procedure.Assuming that sampling is from an absolutely continuous distribution,they showed that ⌢ µ RSS is unbiased for µ and has higher efficiency than ⌢ µ SRS .They established the following bounds for the relative precision (efficiency):1 ≤ RP = Var (⌢ µ SRS )Var ( ⌢ µ RSS ) ≤ m + 1 .2The upper bound is achieved when the underlying distribution is uniform.RSS has many applications in different areas such as biology, medicine andagriculture, (see Al-Saleh et al. (2000), Takahasi and Wakimoto (1968)). Manyauthors evaluated the performance of RSS using real application (see Halls andDell (1966), Evans (1967), Martin et al (1980), Al-Saleh et al (2000), Al-Saleh andAl-Shrafat (2000), Zheng and Al-Saleh (2001) and Al-Saleh and Zheng (2001)).Stokes (1976) examined the potential of RSS for a variety of inferential problemsincluding estimating location and scale parameters, interval estimation, varianceestimation and estimation of correlation coefficient. In 1995, she considered thelocation scale family. An annotated bibliography on RSS was provided by Kaur etal (1995). Double ranked set sampling was considered by Al-Saleh and Al-Kadiri(2000). This was generalized to multistage ranked set sampling by Al-Saleh andAl-Omari (2001). See also Al-Saleh and Samawi (2000).Al-Saleh and Muttlak (1998) made a numerical comparison between theBayes risks of the Bayes estimators obtained based on RSS and SRS for thecase of exponential distribution; no theoretical results were provided. Lavine(1999) has explored different aspects of Bayesian ranked set sampling, examinedthe procedure from a Bayesian point of view and explored some optimality questions.The concept of Bayesian methods along with RSS was studied by Al-Salehet al (2000), who found that for exponential family with conjugate prior, the RSSBayes estimator has smaller Bayes risk than SRS Bayes estimator. In a generalset up, they showed that the SRS Bayes estimator is a weighted average of them m possible RSS plans Bayes estimators. Furthermore, the Bayes risk of RSS

CHARACTERIZATION OF RANKED SET SAMPLING BAYES ESTIMATORS 225Bayes estimator is smaller than the Bayes risk of SRS Bayes estimator, for at leastone plan. The authors have noted that Bayes estimators using RSS have verycomplicated forms even for small sample sizes. Their computations in the simpleexponential case for samples of size 2 and 3 demonstrated this fact. Bayesianestimation of specified parameters under both balanced and generalized RSS wasaccomplished using the Gibbs sampler by Kim and Arnold (1999).The reason that the RSS Bayes estimators have not been popular is that thelikelihood function becomes so complicated, thus making the posterior also verycomplicated, no matter how simple the prior may be. These complications aredue to the fact that there is no minimal sufficient statistic of lower dimensionthan the dimension of the data itself. This lower dimension sufficient statisticusually exists in the case of SRS. In this paper we will use the concept of multipleimputation proposed by Rubin (1978,1987) to obtain a formula that relates theposterior distribution of a parameter θ given the RSS data to that given the fulldata x. Using this formula, it is possible to obtain a characterization of the RSSBayes estimator in terms of SRS Bayes estimator based on the full data. Thisformula facilitates the study of some of the theoretical properties of the estimatoras well as providing clues for approximating their complicated exact forms. Thischaracterization is applied to the case of normal distribution in section 3.2. General Setup and Some Useful ResultsAssume X 1 , X 2 , . . . , X m is a SRS with common density f(x | θ) and anabsolute continuous distribution function F (x | θ). Let Y 1 , Y 2 , . . . , Y m be a RSSfrom this distribution obtained using a full data of m 2 observations,X 11 , . . . , X 1m X (11) = Y 1.X m1,...,Xmm} {{ }X (mm) = Y m} {{ } .full dataRSS.Let Y =∑ mi=1Y imand X = ∑ mi=1X im .

226 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASIt is assumed through out this paper that the judgemental identification ofthe ranks is perfect and with negligible cost. This assumption is essential for mostpublished results on RSS. Under this assumption Y i has the same distribution asX (i) , where X (i) is the i th ordered statistics of a random sample of size m. Also,the squared error loss function is used to compare the estimators.Let π(θ) be a prior of θ, π(θ | y) and π(θ | x) are the posterior density ofθ given y and the posterior density of θ given x, respectively. Throughout thepaper we will be using the following notations:ˆθ SRS (x): The Bayes estimator of θ based on a SRS, X 1 , X 2 , . . . , X m , using anyprior density of θ.ˆθ RSS (y): The Bayes estimator of θ based on a RSS Y 1 , Y 2 , . . . , Y m using any priordensity of θ.ˆθ j SRS (x): The Generalized Bayes estimator of θ based on a SRS X 1, X 2 , . . . , X musing Jeffery prior of θ.ˆθ j RSS (y): The Generalized Bayes estimator of θ based on a RSS Y 1, Y 2 , . . . , Y musing Jeffery prior of θ.ˆθ ∗ SRS (x): The Bayes estimator of θ based on the full data X 11, X 1m , . . . , X mmusing any prior density of θ.ˆθ ∗jSRS (x): The Generalized Bayes estimator of θ based on the full data X 11, . . . ,X 1m , . . . , X mm using Jeffery prior of θ.For more about Bayesian terminology, see Berger (1980).Next, we will state and the following identities that are similar to thoseprovided by Rubin (1978, 1987), for grouped data. The proofs of these identitiesare straightforward and therefore skipped.Identity 1.∫π(θ | y) =−∞xπ(θ | x)m(x | y)dxwhere, m(x | y) = m(x,y)m(y)and∫ ∞∫m(x, y) = π(θ)f(x, y | θ)dθ and m(y) = m(x, y)dx.x

CHARACTERIZATION OF RANKED SET SAMPLING BAYES ESTIMATORS 227This identity says that the posterior density of θ given the RSS data of size mis the expected value of the posterior density of θ, given the SRS data of sizem 2 (full data). The expectation is with respect to the predictive distribution,m(x | y).Identity 2.∫⌢θ RSS (y) =x⌢θ∗SRS(x)m(x | y)dx.This identity says that the RSS Bayes estimator is the expected value of the SRSBayes estimator based on the full data with respect to the predictive distribution.Identity 3.Var (θ | y) = E[Var (θ | X) | y] + Var [ ⌢ θ ∗ SRS(X) | y].Let r( ⌢ θ RSS(Y ), π) = E θ E Y [( ⌢ θ RSS − θ) 2 ] be the Bayes risk of ⌢ θ RSS andr( ⌢ θ ∗ SRS(X), π) = E θ E X [( ⌢ θ ∗ SRS − θ) 2 ] be the Bayes risk of ⌢ θ ∗ SRS.Identity 4.r( ⌢ θ RSS(Y ), π) = r( ⌢ θ ∗ SRS(X), π) + E[Var ( ⌢ θ ∗ SRS(X) | Y )].Thus, we conclude that the Bayes estimator based on RSS of size m can’t bebetter than the Bayes estimator based on SRS of size m 2 (full data).3. Bayes Estimator of the Normal Mean with Conjugate PriorBefore we seek some characterization for the RSS Bayes estimator, ˆθ RSS (y),of the normal mean θ, when θ has the conjugate prior, N(0, 1), we will give someproperties for the generalized RSS Bayes estimator, ⌢ θ J RSS(y). Let Y 1 , . . . , Y m bea RSS from N(θ, 1) then the p.d.f of Y i isg Yi (y i | θ) =m!(i − 1)!(m − i)! [Φ(y i − θ)] i−1 [1 − Φ(y i − θ)] m−i φ(y i − θ)where Φ, φ are the cumulative distribution function and the density function ofa standard normal variable, respectively. Their joint density ism∏g y (y | θ) = g yi (y i | θ).i=1

228 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASAssuming that π(θ) = 1 (non-informative Jeffery prior), then the mean of theposterior distribution of θ given y, is the called the generalized Bayes estimatordenoted by ˆθ J RSS · ⌢θ J SRS is defined similarly. The following properties are easy toverify.Property 1. For a scalar a, let a = (a, . . . , a), and y = (y 1 , . . . , y m ) then⌢Jθ RSS (y + a) = ⌢ θ J RSS (y) + a.Property 2. The risk function of ⌢ θ J RSS is free of θ.Property 3. ⌢ θ J RSS (−y 1, . . . , −y m ) = − ⌢ θ J RSS (y m, . . . , y 1 ).Property 4. ˆθ RSS J is an unbiased estimator of θ, i.e. E(ˆθ RSS J (Y ) | θ) = θ.Now, if X 1 , X 2 , . . . , X m is a SRS from N(θ, 1), then ˆθ SRS is mX1+m, with Bayesrisk= 11+m . Also, ˆθ SRS J is X.Therefore, by Identity 2ˆθ RSS (y) = E(ˆθ SRS ∗ (X) | y) =m2m 2 + 1 E(X∗ | y).Here X ∗ is the average of SRS of size m 2 (full data). Hence,ˆθ RSS (y) =m2m 2 + 1 E(X∗ | y) =m2 ∗Jm 2 E(ˆθ+ 1SRS(X) | y)= m2m 2 + 1 ˆθ RSS J (y). (By Identity 2).Now, based on the last relation we state and prove the following lemma.Lemma 1.(1) r(ˆθ RSS , π) =(2) r(ˆθ RSS , π) ≥1(m 2 + 1) 2 [m4 Var (ˆθ J RSS(Y ) | θ) + 1].0.4805m + 0.5195 + m 3(m 2 + 1) 2 (0.4805m + 0.5195) .(3) 1 ≤ efficiency ≤ (m2 + 1) 2 (0.4805m + 0.5195)(m + 1)(0.4805m + 0.5195 + m 3 ) .where the efficiency = r(ˆθ SRS ,π)r(ˆθ RSS ,π) .

CHARACTERIZATION OF RANKED SET SAMPLING BAYES ESTIMATORS 229Proof.(1) ˆθRSS (y) = m2m 2 + 1 ˆθ RSS J (y) impliesE(ˆθ RSS (Y ) | θ) =m2m 2 + 1 E(ˆθ J RSS(Y ) | θ)= m2m 2 θ. (By Property 4).+ 1Also, Var (ˆθ RSS (Y ) | θ) =Property 2. Hence,( m 2m 2 +1) 2Var (ˆθJ RSS (Y ) | θ), which is free of θ byR(ˆθ RSS (Y ), θ) = Var (ˆθ RSS (Y ) | θ) + (bias) 2( m2 ) 2 θ 2=Var (ˆθJm 2 + 1RSS (Y ) | θ) +(m 2 + 1) 2 .Therefore, the Bayes risk for ˆθ RSS isr(ˆθ RSS (Y ), π) = E θ (R(ˆθ RSS (Y ), θ)) =1(m 2 + 1) 2 [m4 Var (ˆθ RSS J (Y ) | θ) + 1].(2) To find a lower bound for the Bayes risk of ˆθ RSS , it is known by Rao-Cramerinequality that Var (ˆθ j 1RSS(y) | θ) ≥I ∗ (θ) , where I∗ (θ) is the informationnumber in RSS about θ. But I ∗ (θ) = m + m(m − 1)(0.4805), (see Stokes1995), hence[]1m 4r(ˆθ RSS , π) ≥(m 2 + 1) 2 m + m(m − 1)(0.4805) + 1=0.4805m + 0.5195 + m 3(m 2 + 1) 2 (0.4805m + 0.5195) .(3) Efficiency = r(ˆθ SRS , π)r(ˆθ RSS , π) = 1(m + 1)r(ˆθ RSS , π)≤ (m2 + 1) 2 (0.4805m + 0.5195)(m + 1)(0.4805m + 0.5195 + m 3 )(By (2)).To find a lower bound for the efficiency, note that for any estimator of θ, ˆθ,we have by the definition of the Bayes estimator r(ˆθ RSS , π) ≤ r(ˆθ, π). Thus,

230 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASmr(ˆθ RSS , π) ≤ r(m + 1 Y , π)m= E π [Var (m + 1 Y | θ) + (bias)2 ]m1= (m + 1 )2 Var (Y | θ) +(m + 1) 2 ≤ 1(m + 1) 2 [m2 Var (X | θ) + 1]= 1m + 1 .Since Var (Y | θ) ≤ Var (X | θ) = 1 mby Takahasi and Wakimoto (1968), whereY , X are based on the same number of quantifications m.Therefore, efficiency = r(ˆθ SRS ,π)∴ r(ˆθ RSS , π) ≤ 1m + 1 .1=(m+1)r(ˆθ RSS ,π) r(ˆθ RSS ,π)≥ 1 and hence 3 holds.Table 1 shows a numerical comparison between the Bayes risk of ˆθ RSS andits efficiency (with respect to ˆθ SRS ) obtained by simulation work, and thatTable 1.Bayes Risk and Efficiency Obtained by Simulation and that ObtainedUsing the Bounds in (2) and (3), m =set size, r = # of cyclesr(ˆθ RSS , π) Lower bound for Upper bound for Efficiencyr m From simulation r(ˆθ RSS , π) by (2) efficiency by (3) From simulation1 2 0.2502 0.2561 1.3014 1.30443 0.1443 0.1477 1.6928 1.68734 0.0971 0.0942 2.1240 2.07825 0.0644 0.0648 2.5735 2.57152 2 0.1437 0.1458 1.3720 1.37183 0.0794 0.0790 1.8072 1.81224 0.0490 0.0491 2.2648 2.26605 0.0337 0.0333 2.7318 2.69533 2 0.1009 0.1018 1.4028 1.40283 0.0538 0.0540 1.8532 1.85214 0.0339 0.0332 2.3191 2.31075 0.2240 0.0224 2.7912 2.7554

CHARACTERIZATION OF RANKED SET SAMPLING BAYES ESTIMATORS 231obtained using the bounds in (2), (3) of Lemma 1. We can see that the Bayesrisk lower bound values from (2) is very close to its approximated values foundby simulatioin, so, we could use the lower bound as the value for the Bayes riskwithout getting involved into simulation work. Also, most of the efficiency values(from simulation) are within the upper bound of the efficiency found from (3).The few cases, in which the values of the efficiency obtained are higher than theupper bound, were due to sampling variations.4. Approximations of ˆθ RSSAs a useful use of the relation ˆθ RSS (y) =m2m 2 +1 E(X∗ | y), one can think ofdifferent ways of approximating the conditional density of the complete samplegiven the RSS, y. This in turn gives different approximations to ˆθ RSS .As examples, we will discuss two ways of approximating ˆθ RSS .(A) To find ˆθ 1 , as an approximation to ˆθ RSS we reproduce a full data set of sizem 2 using the following two steps:(1) Order the RSS, y 1 , . . . , y m , to get y (1) , . . . , y (m) .(2) Select m values independently from U(y (i) , y (i+1) ) i = 1, . . . , m − 1.This will yield m(m − 1) data points. Using these data points along withthe original m RSS data points as the full data we have:ˆθ RSS (y)=m2m 2 +1 E(X∗ | y) ∼ = ˆθ 1 (y)= 1 [ m+2m 2 (y+1 2 (1) +y (m) )+(1+m)m−1 ∑i=2y (i)].(B) The other approximation ˆθ 2 is found by again ordering the RSS to get y (1) , . . . ,y (m) . Since X i ∼ N(0, 2) (unconditionally), we may reproduce a full data setas follows:Select m values independently from N(0, 2), such that these values are betweeny (i) , y (i+1) ; i = 1, . . . , m−1, i.e. from the density φ( √ x 2)/ √ [2 Φ( y (i+1)√2)−Φ( y ](i)√2) , which is a truncated N(0, 2). This will yield m(m − 1) data pointsthat can be used along with the m RSS data as the full data.Now, theexpected value of each selected observation between y (i) , y (i+1) can be shownto be√2[φ( y (i)√2) − φ( y ](i+1)√2)[Φ( y (i+1)) − Φ( y ] , i = 1, . . . , m − 1.(i)√2)√2

232 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWASThus,ˆθ RSS (y) =m2m 2 +1 E(X∗ | y) ∼ = ˆθ 2 =m(m 2 y+ √ 2+1m−1 ∑i=1[φ( y (i)√2)−φ( y ](i+1) )√2)[Φ( y (i+1)√2)−Φ( y ] .(i)√2)Table 2 shows exact values for ˆθ RSS , obtained using intensive simulation,along with values of the approximations ˆθ 1 , ˆθ2 for different values of θ whenm = 2, 3. It can be seen that the two approximations are very close to ˆθ RSS .Table 2.Simulated vaules of θ, ˆθ 1 , ˆθ 2 , ˆθn θ RSS ˆθRSS ˆθ1 ˆθ22 -2.0553 -1.7684 -1.3709 -1.4530 -1.4482-1.8641-0.4665 -0.1953 -0.2484 -0.2292 -0.2285-0.3777-0.1543 0.4420 0.1977 0.1976 0.19680.51980.1588 1.3029 0.7280 0.7172 0.70610.49020.2298 1.1069 0.2168 0.2110 0.1988-0.57953 -0.2744 1.3300 0.1690 0.1720 0.21010.7726-1.87810.3220 1.0184 0.3897 0.3943 0.38930.3527-0.00564.7351 0.7509 -5.8211 -4.7759 10 −2 -4.2693 10 −210 −2 .0710 10 −2-1.05563 -0.2162 0.8056 -0.2406 -0.2362 -0.16520.3348-2.28620.4424 1.4390 0.5599 0.5491 0.53390.30640.2673

CHARACTERIZATION OF RANKED SET SAMPLING BAYES ESTIMATORS 233The above procedure of studying the properties of RSS Bayes estimators aswell as approximating them, based on their relation to SRS Bayes estimators,can be applied in a similar fashion to other distributions.5. Concluding RemarksRanked set sampling technique is applicable in very specific situations, butwhen applicable, it gives more efficient estimators than simple random sampling.RSS Bayes estimators can be more efficient than the corresponding SRS Bayesestimators. However those estimators have very complicated forms. Using thecharacterization provided in this paper, these estimators can be approximatedand their properties can be investigated.AcknowledgementsWe are grateful to the referees for their constructive and helpful commentsand suggestions which greatly improve the final version of the paper.References[1] M. Fraiwan Al-Saleh and A. I. Al-Omari, Multistage ranked set sampling, Journal of StatisticalPlanning and Inference, to appear.[2] M. Fraiwan Al-Saleh and G. Zheng, Estimation of bivariate characteristics using ranked setsampling. Australian & New Zealand Journal of Statistics, to appear.[3] M. Fraiwan Al-Saleh, K. Al-Shrafat and H. Muttlak, Bayesian estimation using ranked setsampling, Biometrical Journal 42(2000), 1-12.[4] M. Fraiwan Al-Saleh and H. Muttlak, A note on the estimation of the parameter of theexponential distribution using Bayesian RSS. Pakistan Journal of Statistics, 14(1998), 49-56.[5] M. Fraiwan Al-Saleh and K. Al-Shrafat, Estimation of the average milk yield of sheep usingranked set sampling. Environmentrics, 12(2000), 395-399.[6] M. Fraiwan Al-Saleh and H. Samawi, On the efficiency of Monte Carlo methods usingsteady state ranked simulated samples, Communication in Statistics (Simulation and Computation),29(2000), 941-954.[7] J. Berger, Statistical Decision Theory, Springer-Verlag, 1980.[8] M. Evans, Application of ranked set sampling to regeneration surveys in areas direct-seededto long leaf pine. Master Thesis, School of Forestry and Wild Life Management, LouisianaState University, Baton Rouge, Louisiana, 1967.[9] L. Halls and T. Dell, Trial of ranked set sampling for forage yields, Forest Science, 12(1966),22-6.[10] A. Kaur, G. Patil, A. Sinha and C. Tailie, Ranked set sampling: an annotated bibliography,Environmental and Ecological Statistics, 2(1995), 25-45.

234 MOHAMMAD FRAIWAN AL-SALEH AND JAWAHER YOUSEF ABUHAWWAS[11] Y. Kim, B. Arnold, Parameter estimation under generalized ranked set sampling. Statisticsand Probability Letters, 42(1999), 353-360.[12] M. Lavine, The Bayesics of ranked set sampling, Journal of Environmental and EcologicalStatistics, 6(1999), 47-57.[13] W. Martin, T. Sharlk, R. Oderwald and D. Smith, Evaluation of ranked set samplingfor estimating shrub phytomass in application Oak forests publication number FWS-4-80, School of Forestry and Wildlife Resources, Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia, 1980.[14] G. McIntyre, A method for unbiased selective sampling using ranked set, Australian Journalof Agricultural Research. 3(1952), 385-90.[15] D. Rubin, Multiple imputations in sample surveys - a phenomenological Bayesian approachto nonresponse, Proc. Survey Res. Methods Sec. Amer. Statist. Assoc., Washington, 1978.[16] D. Rubin, Multipe imputations for nonresponse in sample surveys and censuses, Wiley, NewYork, 1987.[17] S. Stokes, An investigation of the consequences of ranked set sampling, Ph. D. thesis,Department of Statistics, University of North Carolina, Chapel Hill, North Carolina, 1967.[18] S. Stokes, Parametric ranked set sampling, Annals of Institute of Mathematical Statistics,47(1995), 465-482.[19] K. Takahasi and K. Wakimoto, On unbiased estimates of the population mean based on thesample stratified by means of ordering. Annals of the Institute of Statistical Mathematics,20(1968), 1-31.[20] G. Zheng and M. Fraiwan Al-Saleh, Modified maximum likelihood estimators based on rankedset sampling. The Annals of the Institute of Statistical Mathematics, to appear.Department of Mathematics and Statistics, Sultan Qaboos University, Sultanate of Oman.E-mail: malsaleh@squ.edu.omMathematics Department, Hashemiah University, JORDAN.