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1 This article was downloaded by: [Ferdowsi University of Mashhad] On: 7 June 2010 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Journal of Statistical Computation and Simulation Publication details, including instructions for authors and subscription information: Efficiency of ranked set sampling in entropy estimation and goodness-offit testing for the inverse Gaussian law Mahdi Mahdizadeh a ; N. R. Arghami a a Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran First published on: 25 August 2009 To cite this Article Mahdizadeh, Mahdi and Arghami, N. R.(2010) 'Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law', Journal of Statistical Computation and Simulation, 80: 7, , First published on: 25 August 2009 (ifirst) To link to this Article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 Journal of Statistical Computation and Simulation Vol. 80, No. 7, July 2010, Efficiency of ranked set sampling in entropy estimation and goodness-of-fit testing for the inverse Gaussian law Mahdi Mahdizadeh and N.R. Arghami* Department of Statistics, School of Mathematical Sciences, Ferdowsi University of Mashhad, P.O. Box , Mashhad, Iran (Received 7 July 2008; final version received 23 January 2009 ) When measuring units are expensive or time consuming, while ranking them is relatively easy and inexpensive, it is known that ranked set sampling (RSS) is preferable to simple random sampling (SRS). Many authors have suggested several extensions of RSS. As a variation, Al-Saleh and Al-Kadiri [Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp ] introduced double ranked set sampling (DRSS) and it was extended by Al-Saleh and Al-Omari [Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp ] to multistage ranked set sampling (MSRSS). The entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of information required on the average to determine the value of a (discrete) r.v.. In this work, we discuss entropy estimation in RSS design and aforementioned extensions and compare the results with those in SRS design in terms of bias and root mean square error (RMSE). Motivated by the above observed efficiency, we continue to investigate entropy-based goodness-of-fit test for the inverse Gaussian distribution using RSS. Critical values for some sample sizes determined by means of Monte Carlo simulations are presented for each design. A Monte Carlo power analysis is performed under various alternative hypotheses in order to compare the proposed testing procedure with the existing methods. The results indicate that tests based on RSS and its extensions are superior alternatives to the entropy test based on SRS. Keywords: double and multistage ranked set sampling; imperfect ranking; judgement-ranked set 1. Introduction Ranked set sampling (RSS), in which a large number of units are judgement-ranked with negligible cost and a smaller number of units from the ranked set are actually measured, was first proposed by McIntyre [1] to estimate the population mean. Since then there has been substantial progress in studying this sampling scheme and its extensions. When the variable of interest can be more readily ranked than measured, RSS provides improved statistical inference. We now briefly explain the concept of RSS for completeness. (1) k random samples, each of size k, are drawn from the population. (2) The ith sample (i = 1,...,k) is inspected to identify the unit of (judgement) ith lowest rank. (3) Finally, the k identified units in the (judgement) ranked set are measured. *Corresponding author. arghami_nr@yahoo.com ISSN print/issn online 2010 Taylor & Francis DOI: /

3 762 M. Mahdizadeh and N.R. Arghami The set of measured units then constitutes the ranked set sample of total size k. This process may be repeated l times (i.e. l cycles) to yield a sample of total size n = kl. The resulting ranked set sample is denoted by {X [i]j : i = 1,...,k; j = 1,...,l}, where X [i]j is the unit of ith lowest rank measured from the ith sample in the jth cycle. Note that the n resulting measurements are independently, but not identically distributed. Assuming no judgement error is made, X [i]j is marginally distributed as the ith order statistic of a simple random sample of size k from the same population. RSS has been under vast investigations since its introduction. Patil et al. [2] comprehensively reviewed works done on RSS. However, we will briefly review some of the literature, especially parts relevant to this study. Takahasi and Wakimoto [3] and Dell and Clutter [4] were the first authors who examined theory of RSS mathematically. They showed formally that the mean in RSS is an unbiased estimator of the population mean and is more efficient than the mean in simple random sampling (SRS). Stokes [5] proposed an RSS estimator of the population variance and showed its asymptotic unbiasedness regardless of the presence of errors in ranking. Furthermore, asymptotic efficiency of the estimator, relative to that based on the same number of quantified units in SRS, is greater than unity for any underlying distribution, even if ranking errors occur. MacEachern et al. [6] developed an unbiased estimator of the variance of population based on an RSS and demonstrated that this new estimator is more efficient than that of Stokes. Stokes and Sager [7] studied the properties of the empirical distribution function based on RSS and showed that it is unbiased and has greater precision than that from SRS. Bohn and Wolfe [8] used the RSS empirical distribution function to construct a distribution-free competitor to the Mann Whitney Wilcoxon testing procedure. In all the problems investigated in the literature, RSS results in improved procedures over SRS. It is worth noting that increasing the set size, k, will lead to increased efficiency of RSS over SRS. However, k should be kept small to reduce errors in judgement ranking. Al-Saleh and Al- Kadiri [9] introduced double ranked set sampling (DRSS) technique which improves efficiency of RSS estimator of the population mean for fixed k. Al-Saleh and Al-Omari [10] considered multistage ranked set sampling (MSRSS) as an extension of DRSS. The MSRSS (in one cycle) is described as follows: (1) k r+1 units from the population are randomly identified, where r is the number of stages. (2) The units are then randomly divided into k r 1 sets of k 2 units each. (3) For each set in step (2), use (2) of the RSS procedure described above, to get a (judgement) ranked set of size k. This step gives k r 1 (judgement) ranked sets, each of size k. (4) Without actually measuring the units in the ranked sets, step (3) is applied on the k r 1 ranked sets to gain k r 2 second stage (judgement) ranked sets, of size k each. (5) Without any actual measurement, step (3) is repeated until an rth stage (judgement) ranked set of size k is attained. (6) Finally, the k identified units in step (5) are measured. Similarly, if the process is repeated l cycles to have a sample of size n = kl, the resulting MSRSS will be denoted by {X (r) [i]j : i = 1,...,k; j = 1,...,l}, where r is the number of stages. DRSS is the case that corresponds to r = 2 and if r is set to 1, we have RSS. Entropy of a random variable (r.v.) is a measure of its uncertainty. It is a measure of the amount of (Shannon) information required, on the average, to determine the value of a (discrete) r.v. Information-theoretic measures have served as powerful tools in statistical inference. In using entropy for testing distributional assumptions, the earliest work dates back to Vasicek [11], which used Shannon s maximum entropy characterization result to construct a goodness-of-fit test for normality. Sparked by his work, entropy-based tests of fit were developed for several parametric

4 Journal of Statistical Computation and Simulation 763 models. For instance, Ebrahimi et al. [12], and Grzegorzewski and Wieczorkowski [13] applied the same approach to testing exponentiality. Tests for normality and exponentiality based on RSS data were given in a recent paper by Amini et al.[14]. Recently, Mudholkar and Tian [15] obtained an entropy characterization of the inverse Gaussian distribution and used it to develop an entropybased test of fit for the inverse Gaussian law. In this paper, we will be concerned with the use of RSS and some of its ramifications to improve entropy estimation and goodness-of-fit tests based on entropy. This article is organized as follows. In Section 2, we present entropy estimators, using RSS, DRSS and MSRSS data and compare their performances with estimators in SRS. Section 3 describes goodness-of-fit test procedures for the inverse Gaussian distribution in SRS, RSS, DRSS and MSRSS designs. The critical values of the test statistic estimated by Monte Carlo simulations are tabulated for some sample sizes in each design. In Section 4, a Monte Carlo simulation is carried out to compare the entropy-based tests in different designs in terms of power under several alternative distributions. The effect of imperfect ranking is discussed in Section 5. We state our conclusions in Section Entropy estimation Let X be a continuous r.v. with probability density function f(x), x R. The quantity H(f) = f(x)log f(x)dx (1) is known as Shannon s entropy. The problem of estimating Equation (1) has been considered by many authors. Substitution of f in the formula for H(f) by one of its estimates gives a method of entropy estimation; the resulting estimators obtained in this manner are asymptotically unbiased and consistent under some assumptions. Vasicek [11] proposed an estimator of the entropy of univariate distributions based on spacings. His estimate was based on the fact that Equation (1) can be expressed as H(f) = 1 0 ( ) d log dp F 1 (p) dp. (2) The estimate was constructed by replacing distribution function F by the empirical distribution function. The derivative of F 1 (p) at p = i/n is then estimated by (x (i+m) x (i m) )n/(2m). Let X (1),...,X (n) be the order statistics of a random sample from distribution F. The estimator is given by V m,n = 1 n n i=1 { n } log 2m (X (i+m) X (i m) ), (3) where m (called window size) is a positive integer (m n/2), X (i) = X (1) for i<1, and X (i) = X (n) for i>n.

5 764 M. Mahdizadeh and N.R. Arghami Since entropy estimator (3) is a function of order statistics, entropy estimation using RSS data involves ordering the ranked set sample. The natural way is to pool units in all cycles and construct the estimator based on ordered pooled sample. The resulting estimator is given by V (r) m,n = 1 n n i=1 { n } log 2m (X(r) (i+m) X(r) (i m) ), (4) where X (r) (a) is the ath-order statistic of the pooled MSRSS (a = 1,...,n)and r shows the number of stages. Here the subscript k (the set size) has been suppressed for convenience. We use V m,n (0) to denote V m,n in Equation (3). A simulation study was conducted to see the behaviour of the proposed estimators. For each sample size (10, 20, 30), 10,000 samples were generated in SRS, RSS and DRSS designs, the estimators V m,n (r) for r = 0, 1, 2 were computed and their biases and RMSEs were determined. Since the set size k should be small, k = 10 and l = 1, 2, 3 were used to draw samples of sizes n = 10, 20, 30 in RSS and DRSS designs, respectively. This setup applies to all the tables. Here n is the total sample size. For instance, if l = 2, the number of units that are actually measured are 20, 10 in each cycle. Tables 1 3 contain simulated biases and RMSEs for the uniform, exponential and normal distributions at selected sample sizes for each design. It is observed that Table 1. Monte Carlo biases and RMSEs of V (r) m,n for the uniform (0,1) distribution, H(f) = 0. SRS RSS DRSS n m Bias RMSE Bias RMSE Bias RMSE

6 Journal of Statistical Computation and Simulation 765 Table 2. Monte Carlo biases and RMSEs of V (r) m,n for the exponential distribution with mean one, H(f) = 1. SRS RSS DRSS n m Bias RMSE Bias RMSE Bias RMSE for all distributions and sample sizes, RSS and DRSS designs have lower biases and RMSEs than SRS. It is seen that using DRSS design improves the results substantially, as compared with SRS. 3. Testing procedure Many well-known distributions can be characterized in terms of entropy and the use of such entropy characterization results is an appealing approach for constructing goodness-of-fit tests. As noted by Mudholkar and Tian [15], the inverse Gaussian distribution is not included in this category, since a direct characterization of that as a maximum entropy distribution is not available. Hence, they presented the next interesting result (Corollary 2) and used it to develop an entropybased test of fit for the inverse Gaussian distribution. Corollary 1 Suppose X has an inverse Gaussian IG(μ, λ) distribution with the p.d.f. ( ) λ 1/2 { f(x)= exp λ } (x 2πx 3 2μ 2 μ)2, x

7 766 M. Mahdizadeh and N.R. Arghami Table 3. Monte Carlo biases and RMSEs of V (r) m,n for the standard normal distribution, H(f) = SRS RSS DRSS n m Bias RMSE Bias RMSE Bias RMSE and let Y = 1/ X. Then the entropy of Y is given by H(f Y ) = log ( ξ ) 2πe, 2 where ξ 2 = 1/λ = E(Y 2 ) 1/E(Y 2 ). Corollary 2 The r.v. X with inverse Gaussian distribution is characterized by the property that 1/ X attains the maximum entropy among all nonnegative, absolutely continuous r.v. s Y with a given value at E(Y 2 ) 1/E(Y 2 ). Suppose x (1) x (n) are the observed order statistics of a random sample according to the density f(x) and we are interested in testing whether this random sample is from an inverse Gaussian population. They proposed rejecting the composite null hypothesis H 0 : X IG(μ, λ) if K m,n (f Y ) = exp{v m,n (f Y )}/(w/2) K m,n,α (f Y ), (5)

8 Journal of Statistical Computation and Simulation 767 where V m,n (f Y ) = 1 n n i=1 { n } log 2m (y (i+m) y (i m) ), (6) is the estimate of the entropy of Y = 1/ X, where y (i) = 1/ x (n i+1) (i = 1,...,n), w 2 is UMVU estimate of ξ 2 given by w 2 = = n (1/x i 1/ x)/(n 1) i=1 ( n n yi 2 /(n 1) n2 i=1 i=1 y 2 i ) 1 / (n 1), (7) and Km,n,α (f Y ) is the 100α percentile of the null distribution of K m,n. It is difficult to derive the null distribution of K m,n analytically, so critical values of the test statistic were determined by means of Monte Carlo simulations. The generation of random samples from the the inverse Gaussian distribution is easily done by R statistical software by installing SuppDists package 1 version written by Bob Wheeler and loading it and then using the command rinvgauss(n,mu,lambda), where mu and lambda are the parameters of the distribution, and n is the sample size. In order to carry out the simulations for the present paper, 10,000 samples were generated from the IG(1, 1) distribution in SRS, RSS and DRSS designs for each sample size (10, 20, 30). Then, the estimators V m,n (r) for r = 0, 1, 2 were computed, placed in Equation (5) and Km,n,0.05 was determined. In RSS and DRSS designs, we treat data as if they were collected by SRS design to estimate w 2. Table 4 contains 0.05 critical points for the test statistic in different designs. To calculate the test statistic, the window size m corresponding to a given sample size must be selected in advance. In entropy estimation based on spacings, choosing optimal m for given n is still an open problem. For each n, the window size having largest critical value tends to yield greater power. For sample sizes 10, 20 and 30, window sizes producing the maximum critical Table critical points for the test statistic. n m SRS RSS DRSS n m SRS RSS DRSS

9 768 M. Mahdizadeh and N.R. Arghami values of the test statistic are found respectively to be 3, 3 and 4 in all designs. These optimal window sizes, denoted by m, will be used in the next section to study the effect on power properties of the number of stages in MSRSS. A look at Tables 1 3 shows that for each n, the value of m having smallest bias and RMSE, say m, is nearly identical for all distributions. Sample sizes 10, 20 and 30 were used for which m is approximately 3. Since one of the components of the test statistic is an entropy estimator, one would expect to see the window size having the largest critical value, denoted by m, coincides with the corresponding m. Simulations show that, for each sample size, m and m are in good agreement. 4. Power comparison In this section, we investigate the performances of the goodness-of-fit tests under different designs and against the alternatives (A) exponential(1), (B) uniform(0,1), (C) Weibull(2, 1) (with shape parameter 2 and scale parameter 1), (D) lognormal(0, 2) (with mean e 2 and standard error e 2 e 4 1), (E) beta(2, 2) and (F) beta(5, 2), using Monte Carlo simulations. A variety of functional forms are covered by choosing these alternatives; we note that (A), (C) and (D) are right-skewed, (B) and (E) are symmetric and (F) is left-skewed. Table 5. Power comparison for the entropy test at the significance level α = 0.05 under several alternative distributions. Exponential Uniform n m SRS RSS DRSS SRS RSS DRSS

10 Journal of Statistical Computation and Simulation 769 For each alternative, 10,000 samples of sizes 10, 20 and 30 were generated under each design and the power of each test was estimated by the percentages of samples entering the rejection region. Tables 5 7 present the estimated powers of the tests. The results indicate that improvement in entropy estimation using RSS data is transmitted to entropy-based tests giving greater power, as one expects intuitively. Generally, one can say that regardless of the alternative distribution and sample size, RSS and DRSS designs have higher power than SRS and powers in DRSS design dominate those in RSS. The general trend of superiority of DRSS over RSS is sometimes violated. This happens when the power in SRS design is less than 0.15, which corresponds to non-optimal values of m near or equal to n/2. However, it is noteworthy that against all alternatives and for each n, maximum power occurs either for optimal m or for one of its neighboring values. Such a strategy was employed by Ebrahimi et al. [12] in testing exponentiality. Since DRSS design leads to higher power when compared with RSS, it is tempting, although computationally intensive, to see if substantial increases in power would continue if the number of stages is increased to more than two. To respond to this curiosity, Tables 4 7 were produced using MSRSS with r = 3, 4. Since each further stage in MSRSS considerably prolongs the implementation time, we generated 1000 samples instead of 10,000 in computing critical values and powers. Also, results were reported just in the case of optimal m for a given n, not for all values of Table 6. Power comparison for the entropy test at the significance level α = 0.05 under several alternative distributions. Weibull(2, 1) Lognormal(0, 2) n m SRS RSS DRSS SRS RSS DRSS

11 770 M. Mahdizadeh and N.R. Arghami Table 7. Power comparison for the entropy test at the significance level α = 0.05 under several alternative distributions. Beta(2,2) Beta(5,2) n m SRS RSS DRSS SRS RSS DRSS Table critical points of the test statistic under MSRSS designs. Stage number n(m ) r = 2 r = 3 r = 4 10(3) (3) (4) m such that m n/2. Table 8 contains 0.05 critical points for the test statistic and Table 9 gives the powers in different designs. In both the tables, DRSS results, extracted from previous Tables, were included for the sake of comparison. Table 9 shows that as the number of stages increases, power is slightly improved. It is observed that powers for r = 3, 4 are not markedly greater than when r = 2, and thus DRSS, which needs moderate sampling effort, seems to be satisfactory for practical purposes. Therefore, there does not seem to be a need for expanding the sampling plan to MSRSS.

12 Journal of Statistical Computation and Simulation 771 Table 9. Power comparison for the entropy test at the significance level α = 0.05 against several alternative distributions under MSRSS designs. Alternative n(m ) r E(1) U(0,1) W(2,1) LN(0,2) B(2,2) B(5,2) 10(3) (3) (4) Effect of imperfect ranking When an inference procedure based on RSS is proposed, its performance is initially evaluated under perfect ranking assumption, say model A. This means that the judgement-rank order of the units do match with their true numerical orders. Since true orders are unknown unless all units are measured, imperfect ranking is unavoidable. Thus, to complete the evaluation process, one has to show, regardless of the ranking quality, that the procedure remains valid. In extreme cases, ranking may be so poor to yield a simple random sample. Hence, RSS is at least as efficient as SRS, intuitively. Some authors have addressed this issue in specific problems. Dell and Clutter [4] showed that the sample mean of RSS is an unbiased estimator of the population mean even in the presence of ranking errors and has smaller variance than the usual estimator based on SRS with the same sample size. But the variance of the estimator with ranking errors will be larger than the variance of the estimator with perfect ranking, and less than or equal to the variance of the estimator based on SRS. Bohn and Wolfe [16] showed that inference based on Mann Whitney Wilcoxon statistic using RSS is more efficient than SRS, even if perfect ranking is violated. We describe a model for imperfect ranking introduced by Bohn and Wolfe [16]. Let p ij denote the probability that the unit that actually has numerical rank i is judgement-ranked as the jth order statistic. Consider situation in which the unit of true rank i is judgement-ranked as one of its neighbouring order statistics with equal probability, i.e. for i = 2,...,k 1 p ij = { 0.5, j = i 1,i+ 1, 0, j = i and for i = 1,k, p 1j = { 0.5, j = 1, 0.5, j = 2

13 772 M. Mahdizadeh and N.R. Arghami and p kj = { 0.5, j = k, 0.5, j = k 1, respectively, where k is the set size of each ranked sample. This model will be referred to as B. Tables 10 and 11, respectively, present simulated biases and RMSEs of V m,n (1), and the power of the test in RSS design computed under model B along with the corresponding results under model A, extracted from Tables 1 3 and 5 7. It is seen that assuming model B, absolute values of biases, and RMSEs are slightly higher than those of model A. Also powers under model B are comparable with those in model A (and sometimes higher). The fact that ranking errors can sometimes improve the efficiency has also been reported by Pressnel and Bohn [17]. The results suggest that our procedure is fairly robust against violations of perfect ranking assumption. It was shown by Al-Saleh and Al-Kadiri [9] that ranking in the second stage of DRSS is easier than that in the first stage of SRS; it is even easier in rth stage of MSRSS. This property ensures us that DRSS and MSRSS designs are at least as robust as RSS against imperfect ranking. It should be noted that more precise assessment of the performance of statistical methods based on RSS is only possible through developing plausible models for imperfect ranking. Up to the Table 10. Monte Carlo biases and RMSEs of V (1) m,n for the uniform, exponential and normal distributions. U(0,1) E(1) N(0,1) n m Model Bias RMSE Bias RMSE Bias RMSE 10 1 A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B

14 Journal of Statistical Computation and Simulation 773 Table 11. Power comparison for the entropy test at the significance level α = 0.05 under several alternative distributions in RSS design. Alternative n m Model E(1) U(0,1) W(2,1) LN(0,2) B(2,2) B(5,2) 10 1 A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B authors knowledge, such models have not been thoroughly examined and further research in that area is required. 6. Conclusion In this article, RSS and some of its modifications were utilized in two contexts: entropy estimation and goodness-of-fit testing for the inverse Gaussian distribution based on its maximum entropy characterization result. Monte Carlo simulations were used to assess the performance of proposed RSS-based procedures in both contexts. Simulation results report that accuracy of entropy estimation and power of the tests based on RSS and its extensions to DRSS and MSRSS designs are higher than the existing methods based on SRS. The results of this research indicate that there may be potential improvements on many other statistical methods, by using RSS design and its extensions. Exploring such areas and finding newer applications of RSS technique will be worth the effort. Acknowledgements Partial support from Ordered and Spatial Data Center of Excellence of Ferdowsi University of Mashhad"is acknowledged.

15 774 M. Mahdizadeh and N.R. Arghami Note 1. It is freely available at References [1] G.A. McIntyre, A method of unbiased selective sampling using ranked sets, Austral. J. Agric. Res. 3 (1952), pp [2] G.P. Patil, A.K. Sinha, and C. Taillie, Ranked set sampling: a bibliography, Environ. Ecol. Statist. 6 (1999), pp [3] K. Takahasi and K. Wakimoto, On unbiased estimates of the population mean based on the sample stratified by means of ordering, Ann. Inst. Statist. Math. 21 (1968), pp [4] T.R. Dell and J.L. Clutter, Ranked set sampling theory with order statistics background, Biometrics 28 (1972), pp [5] S.L. Stokes, Estimation of variance using judgement ordered ranked set samples, Biometrics 36 (1980), pp [6] S.N. MacEachren, O. Ozturk, D.A. Wolfe, and G.V. Stark, A new ranked set sample estimator of variance, J. R. Statist. Soc. B. 64 (2000), [7] S.L. Stokes and T.W. Sager, Characterization of a ranked-set sample with application to estimating distribution function, J. Amer. Statist. Assoc. 83 (1988), pp [8] L.L. Bohn and D.A. Wolfe, Non-parametric two-sample procedures for ranked-set samples data, J. Amer. Statist. Assoc. 87 (1992), pp [9] M.F. Al-Saleh and M. Al-Kadiri, Double ranked set sampling, Statist. Probab. Lett. 48 (2000), pp [10] M.F. Al-Saleh and A.E. Al-Omari, Multistage ranked set sampling, J. Statist. Plann. Inference 102 (2002), pp [11] O. Vasicek, A test of normality based on sample entropy, J. R. Statist. Soc. B. 38 (1976), pp [12] N. Ebrahimi, M. Habibullah, and E.S. Soofi, Testing exponentiality based on Kullback Leibler information, J. R. Statist. Soc. B. 54 (1992), [13] P. Grzegorzewski and R. Wieczorkowski, Entropy based goodness-of-fit test for exponentiality. Commun. Statist.- Theory Methods 28 (1999), pp [14] M. Amini, M. Mahdizadeh, and N.R. Arghami, Improved estimator of the entropy and goodness-of-fit tests in ranked set sampling, (2008) (submitted). [15] G.S. Mudholkar and L. Tian, An entropy characterization of the inverse Gaussian distribution and related goodnessof-fit test. J. Statist. Plann. Inference 102 (2002), [16] L.L. Bohn and D.A. Wolfe, The effect of imperfect judgement rankings on properties of procedures based on the ranked-set samples analog of the Mann Whitney Wilcoxon statistic, J. Amer. Statist. Assoc. 89 (1994), [17] B. Presnell and L.L. Bohn, U-statistics and imperfect ranking, J. Nonpar. Statist. 10 (1999), pp