An improved ranked set two-sample mann-whitney-wilcoxon test

The Canadian Journal of Statistics Vol. 28, No. 2, 2000, Pages ???-??? La revue canadienne de statistique An improved ranked set two-sample Mann-Whitney-Wilcoxon test Ömer ÖZTÜRK and Douglas A. WOLFE Key words and phrases: Design, unequal allocation, Pitman efficiency, nonparametric testing, mode, information, Mann-Whitney-Wilcoxon test. Mathematics subject classification codes (1991): 62G30; 62H10. ABSTRACT The authors present an improved ranked set two-sample Mann-Whitney-Wilcoxon test for a location shift between samples from two distributions F and G. They define a function that measures the amount of information provided by each observation from the two samples, given the actual joint ranking of all the units in a set. This information function is used as a guide for improving the Pitman efficacy of the Mann-Whitney-Wilcoxon test. When the underlying distributions are symmetric, observations at their mode(s) must be quantified in order to gain efficiency. Analogous results are provided for asymmetric distributions. RÉSUMÉ Les auteurs décrivent une adaptation de la statistique de Mann-Whitney-Wilcoxon permettant de tester si deux lois F et G sont identiques à un paramètre de localisation près, lorsque chacun des deux échantillons est composé de certaines statistiques d’ordre d’un ensemble aléatoire de données classées subjectivement. Ils définissent aussi la quantité d’information associée à chaque observation des deux lois, étant donné les classements subjectifs préalables. Ils montrent alors qu’un choix judicieux du sous-ensemble des données qui seront mesurées objectivement permet d’accroı̂tre l’efficacité du test au sens de Pitman. Lorsque les lois sous-jacentes sont symétriques, c’est en incluant leur(s) mode(s) dans ce sous-ensemble que l’on réalise des gains. Le cas des lois asymétriques est aussi traité. 1. INTRODUCTION In situations where measurements are costly and/or difficult to obtain but ranking of the potential sample data is relatively easy, the use of statistical methods based on a ranked set sampling (RSS) approach can lead to substantial improvement over analogous methods associated with simple random sampling (SRS) schemes. This ranked set sampling approach has attracted considerable attention 1 in the recent literature, with principal initial interest being driven by environmental and agricultural issues, where it is clear that pre-sampling judgement rankings can be quite inexpensive relative to the cost of detailed measurement of many quantities of interest. For example, preliminary supporting data can be easily obtained from a contaminated site and analyzed both quickly and inexpensively before final decisions are reached as to where and how to obtain the specific measurement(s) of interest, such as lead or hazardous material content of the site. A similar approach can be taken when performing standard gasoline octane checks at stations, where preliminary screening at the gasoline pumps can lead to an improved set of ranked set gasoline samples to carry back to the laboratory for more detailed (and costly) analyses. Agricultural settings where such an approach can be used effectively include predicting crop yields or lumber content via preliminary non-destructive measurements (satellite observations or concomitant variables, for example). In RSS, m2 units are drawn from an infinite population. These units are partitioned into m sets each having m units. The units in each set are then judgement ranked via some means other than actual measurement. From the ith set, i = 1, . . . , m, the observation having the ith judgement rank is quantified while other observations are returned to the population. This process is called a cycle in ranked set sampling. The random sample resulting from a cycle contains m independent order statistics. In practice, this entire process is repeated for n cycles to yield n × m quantified observations. Two-sample ranked set samples can be constructed in a way similar to that for the one-sample case. For the two-sample setting, we quantify n × q order statistics, X(i)j , i = 1, . . . , q, j = 1, . . . , n, from the X−sample distribution F and m × p order statistics, Y(i)j , i = 1, . . . , p, j = 1, . . . , m, from the Y −sample distribution G, where F and G are arbitrary, continuous probability models. Throughout the paper, the observations X(i)j , i = 1, . . . , q, j = 1, . . . , n, and Y(r)ℓ , r = 1, . . . , p, ℓ = 1, . . . , m, will be called two-sample standard ranked set samples (SRSS). Most of the initial research efforts in ranked set sampling have concentrated on parametric and nonparametric estimation and testing procedures for the oneand two-sample settings. See, for example, McIntyre (1952), Takahasi & Wakimoto (1968), Dell & Clutter (1972), Halls & Dell (1966), Stokes (1977), Stokes & Sager (1988), Bohn & Wolfe (1992, 1994), Hettmansperger (1995), Bohn (1998), Koti & Babu (1996), and Öztürk (1999a,b). Recently, several researchers have expressed interest in the appropriate allocation of order statistics within a ranked set sample. Kaur et al. (1997) studied the effects of unequal allocation on estimation of the population mean when the underlying distribution is skewed or symmetric. They provided “near” optimality results based on skewness, kurtosis and/or the coefficient of variation. In the parametric set up, Bhoj (1997) constructed an unbiased estimator of the population mean by using a linear combination of the order statistics from the same set in a ranked set sampling environment. Öztürk (1999a) and Öztürk & Wolfe (1998a,b, 1999) introduced a design concept for determining the appropriate allocation of the order statistics for the sign and signed rank test statistics. They showed that for all symmetric distributions the best design among all possible allocation procedures is the one that quantifies only the middle observation(s). Recently, Öztürk & Wolfe (1998a) developed an information criterion to improve the asymptotic Pitman efficiency of a one-sample nonparametric test. In this paper, we extend the same idea to settings where the two-sample ranked set sample MannWhitney-Wilcoxon test is appropriate. This extension to the two-sample problem is not trivial since the class of all possible allocation procedures is very rich and the 2 Pitman efficacy factor for the test is not analytically available for maximization. Therefore, we adopt a two-step procedure to improve the ranked set sample MannWhitney-Wilcoxon test. In Section 2, as a first step we provide the asymptotic Pitman efficiency of the ranked set sample Mann-Whitney-Wilcoxon test for an arbitrary design. We show that the expression of the Pitman efficiency does not permit an analytical maximization with respect to the choice of ranked set sampling design. In order to provide a feasible search algorithm, Section 3 introduces a minimum variance optimality criterion for allocation of the order statistics in a two-sample ranked set sample and construction of minimum variance optimal designs that minimize this criterion function. It is shown that the minimum variance optimal designs are functions only of the modes of the underlying distributions. In Section 4, we use this optimality criterion to aid our search for an improved ranked set two-sample Mann-Whitney-Wilcoxon test. We numerically maximize the Pitman efficiency by using the variance-optimality criterion as a guide. This is a manageable task since the variance-optimal guided search produces a much smaller class than the class of all possible designs. For example, using the variance-optimality criterion for unimodal distributions, the Pitman efficiency need only be numerically maximized within the class of all designs that contain single order statistics from each of the X- and Y -sample distributions. This restricts our search to only pq different designs. All proofs are provided in appendix. 2. RANKED SET SAMPLE MANN-WHITNEY-WILCOXON TEST Let Dts = {d1 , . . . , dt ; b1 , . . . , bs }, t ∈ {1, . . . , q}, s ∈ {1, . . . , p}, be the set of integers that contains the ranks of the observations to be quantified in each cycle from the X− and Y −sample distributions, respectively. For example, if D = {1, 3, 5; 1, 5} and the set sizes are p = 5 and q = 5, we would quantify the two extreme and one middle observations in the X−sample and the two extremes in the Y −sample in each cycle. Throughout this paper the set Dts will be called the design and we reserve DSRSS = {1, . . . , q; 1, . . . , p} for the standard ranked set sample. In order to preserve the U -statistic nature of the Mann-Whitney-Wilcoxon statistic, we quantify equal numbers of observations at each design point (n for the X-sample and m for the Y - sample). Thus, our two-sample ranked set sample is X(di )j , j = 1, . . . , n; i = 1, . . . , t, and Y(bu )v , v = 1, . . . , m; u = 1, . . . , s. For each j = 1, . . . , n and v = 1, . . . , m, let X j = (X(d1 )j , . . . , X(dt )j ) and Y v = (Y(b1 )v , . . . , Y(bs )v ). Then the two-sample ranked set sample with design Dts is given by X j , j = 1, . . . , n and Y v , v = 1, . . . , m. Now let   q−1 F (x)di −1 {1 − F (x)}q−di f (x)dx f(di ) (x) = q di − 1 and g(bi ) (y) = p   p−1 G(y)bi −1 {1 − G(y)}p−bi g(y)dy. bi − 1 Then it follows that X 1 , . . . , X n are iid random vectors from the joint distribution h1 (x) = t Y f(di ) (xi ). i=1 3 Similarly, Y 1 , . . . , Y m are iid random vectors from the joint distribution h2 (y) = s Y g(bi ) (yi ). i=1 Moreover, the X’s and Y ’s are mutually independent. Let U (Dts ) = s m X t X n X X j=1 i=1 v=1 u=1 δ(X(di )j − Y(bu )v ), where δ(z) = 0, 1 according as z > 0 or z ≤ 0. In this equation, U (Dts ) is a multivariate two-sample U -statistic estimator of degree (1, 1) on the random vectors X i , Y j , i = 1, . . . , n and j = 1, . . . , m, for the parameter s t X X γ(Dts ) = i=1 u=1 P (X(di )1 − Y(bu )1 ≤ 0). The associated kernel function for this U -statistic estimator is wDts (X 1 , Y 1 ) = s t X X i=1 u=1 δ(X(di )1 − Y(bu )1 ). Let G(y) = F (y − ∆) with −∞ < ∆ < ∞. We now use U (Dts ) to test H0 : ∆ = 0 in favour of H1 : ∆ > 0 by rejecting H0 for large values of U (Dts ). We first look at the null distribution of U (Dts ). √ Theorem 1. Let K = n + m, ǫ = limK→∞ n/K and σ02 (Dts ) = Var{ KU (Dts )/ (nm)}. If 0 < ǫ < 1 and σ02 (Dts ) > 0, then under the null hypothesis ∆ = 0, √ K{Ū (Dts ) − γ(Dts )} has a normal distribution with mean zero and variance σ02 (Dts ), where Ū (Dts ) = U (Dts )/(nm), ξ1,0 (Dts ) ξ0,1 (Dts ) + , ǫ 1−ǫ σ02 (Dts ) = ξ1,0 (Dts ) = τ1 (Dts ) + τ2 (Dts ), ξ0,1 (Dts ) = τ3 (Dts ) + τ4 (Dts ), with τ1 (Dts ) = s t X X i=1 u=1 τ2 (Dts ) =  bX u −1 u −1 bX   j=0 v=0 t XX X i=1 −q 2  q
p
p
q−1 di −1 j (j + v + di )  v
2p+q j+v+di  q q − 1 1≤r6=ℓ≤s di − 1 q−1 di − 12  bX r −1 bX ℓ −1 j=0 u=0 −  u −1 bX  j=0 p j  bX r −1 bX ℓ −1 j=0
p
u 2p+q (j + u + di ) j+u+d u=0 i 

p j p u (j + di )(u + di ) 4 
p
2  q  j
p+q   , (j + di ) j+d i q−1 di −1 p+q j+di
p+q u+di
 ,
 q q s t X X X X τ3 (Dts ) =  i=1 u=1 j=di v=di p p−1 bu −1
q
q
j (j + v + bu ) and τ4 (Dts ) = v
2q+p j+v+bu −  q X  j=di 
q
2   j
p+q   (j + bu ) j+b p p−1 bu −1 u 

q q q q  p−1 X X c h
p 2q+p 1≤i6=j≤t  bu − 1 (c + h + bu ) c+h+b u=1 c=di h=dj u 

 X q q q q  X p−1 c  h   −p2 . p+q  bu − 12 (j + b )(h + b ) p+q s XX X c=di h=dj u u j+bu h+bu This asymptotic result allows us to carry out an approximate size α test for any arbitrary design Dts . In order to compare the performance of the design Dts and the standard two-sample ranked set sample analogue of the Mann-WhitneyWilcoxon test studied by Bohn & Wolfe (1992, 1994), with design DSRSS , we investigate the Pitman asymptotic relative efficiency, defined as ARE(Dts ; DSRSS ) = ef f (Dts ) , ef f (DSRSS ) where ef f (Dts ) is the efficacy factor for the Mann-Whitney-Wilcoxon test statistic U (Dts ) based on the two-sample ranked set sampling design Dts . The efficacy factor ef f (Dts ) is given by {µ′Dts (0)}2 2 (D ) , N →∞ σ0 ts ef f (Dts ) = lim (1) ∂ where µ′Dts (0) = ∂∆ E∆ Ū (Dts )|∆=0 . Care needs to be taken in order to have a meaningful comparison of the designs Dts and DSRSS . In the derivation of the asymptotic results for Ū (Dts ), the design sizes t and s are kept fixed, while the cycle sizes (m and n) go to infinity. Therefore, the efficiency comparison between designs Dts and DSRSS makes sense only if (t, s) and (q, p) are comparable. Let t∗ = q/t and s∗ = p/s and assume that t∗ and s∗ are integers. Now we consider a two-sample ranked set sample generated by design Dts in such a way that X di = {X(di )1 , . . . , X(di )t∗ }, i = 1, . . . , t, is Qt∗ a t∗ -dimensional random variable from the joint distribution j=1 F(di ) (xj ) and Y bi = {Y(bi )1 , . . . , Y(bi )s∗ }, i = 1, . . . , s, is an s∗ -dimensional random variable from Q∗ ∗ be the design for this multivariate the joint distribution sj=1 F(bi ) (xj −∆). Let Dts setting. ∗ , the Pitman efficacy is Theorem 2.For fixed, but arbitrary, design Dts R Pt Ps t∗ s∗ ǫ(1 − ǫ){ i=1 j=1 f(di ) (y)f(bj ) (y)dy}2 ∗ ef f (Dts ) = s∗ (1 − ǫ)ξ1,0 (Dts ) + t∗ ǫξ1,0 (Dts ) ∗ with respect to the and the Pitman asymptotic relative efficiency of the design Dts two-sample ranked set sample (DSRSS ) is ∗ ; DSRSS ) = TF (t, s, p, q) ARE(Dts (1 − ǫ)ξ1,0 (DSRSS ) + ǫξ0,1 (DSRSS ) , (1 − ǫ)s∗ ξ1,0 (Dts ) + t∗ ǫξ0,1 (Dts ) 5 (2) where TF (t, s, p, q) = t∗ s ∗ { Pt i=1 Ps j=1 R p2 q 2 { R f(di ) (y)f(bj ) (y)dy}2 f 2 (y)dy}2 . Equation (2) does not, in general, permit an analytical solution to maximize ∗ ∗ ARE(Dts ; DSRSS ) with respect to Dts . Note that the class of all possible designs is very rich for large q and p as it contains (2q − 1) × (2p − 1) different designs. ∗ , DSRSS ) over this entire class does not Thus numerical maximization of ARE(Dts appear to be attainable. Therefore, we need to develop a reasonable guided search algorithm to achieve an improvement in the asymptotic Pitman efficiency of the ranked set sample Mann-Whitney-Wilcoxon test. In Section 3 we propose the use of an information function to provide one such search algorithm. 3. INFORMATION FUNCTION Our interest lies in finding a design Dts that, in some sense, contains the maximum amount of information in a two-sample ranked set sample. Then we use this optimal design and associated information function to guide us in our search for an improved ranked set sample Mann-Whitney-Wilcoxon test. The optimality of this design depends, in general, on the type of inference, the underlying distributions and the nonparametric procedure of interest. We expect, by the nonparametric nature of the ranked set sample Mann-Whitney-Wilcoxon test, only a weak dependence on the general shapes of the underlying distributions. As in the one-sample problem (Öztürk & Wolfe 1998a), we define an information function to quantify the amount of information that can be extracted using twosample ranked set sample procedures with unequal allocation from the X and Y distributions, given the judgement ranking of the entire set. To construct such a ranked set sample, we draw N × q units from the X distribution F and M × p units from the Y distribution G. These units are partitioned into N different sets each containing q units for the X−samples and M different sets each containing p units in Y −samples. Units in each set are then judgement ranked. For a given design Dqp , based on this ranking information we quantify the dth i X−sample order statistics Y −sample order statistics in ri sets, i = 1, . . . , p, q, and the bth in ci sets, i = 1, . . . ,P i Pp q in such a way that i=1 ci = N and i=1 ri = M . Then our two-sample ranked set sample is X(di )j , j = 1, . . . , ci ; i = 1, . . . , q, and Y(bt )s , s = 1, . . . , rt ; t = 1, . . . , p. The information that design Dqp carries in this setting is measured as   √ p X q X rt ci X   X q+p T (Dqp ) = Var p (X(di )j − Y(bt )s )   N M (N + M ) i=1 j=1 t=1 s=1 = q X i=1 (1 − 2 ǫ)ai σ(d i :q) ǫ1 with the constraints  Pq Ppi=1 ai = 1, j=1 vj = 1, + p X j=1 2 ǫvj τ(b j :p) 1 − ǫ1 (3) ai ≥ 0, i = 1, . . . , q, vj ≥ 0, j = 1, . . . , p, 2 2 where ai = ci /N, vj = rj /M , ǫ1 = q/(p + q), ǫ = N/(N + M ) and σ(d and τ(b i :q) j :p) √ √ are the variances of qX(di )1 and pY(bj )1 , respectively. Let Dvar be the varianceoptimal design that minimizes T (Dqp ) in equation (3). Note that information in a 6 two-sample ranked set sample is expressed in terms of the magnitude of the total variation in the sample. In order to maximize the sample information, one needs to minimize the total variation with respect to design Dts , where t ∈ {1, . . . , q} and s ∈ {1, . . . , p}. Thus, the design Dts that produces the smallest value of T (Dts ) can be used to guide us in our search to uncover the underlying structure for an improved Mann-Whitney-Wilcoxon test. At this point we need to emphasize two points. First the coefficients ai and vj are functions of the design Dqp . Thus the choices of ai and vj will be determined by the optimal design and ai and vj will both be zero if the corresponding di and bj are not in Dvar . The second point is that the constraints in equation (3) ensure that at least one observation is quantified in each of the X− and Y −samples. We note that we cannot infer that Dvar will maximize the Pitman asymptotic efficiency of the Mann-Whitney-Wilcoxon test among all (2p − 1) × (2q − 1) designs. However, we will show in Section 4 that the use of Dvar does lead to an improvement for the standard ranked set sample Mann-Whitney-Wilcoxon test. We require the following assumptions: A1: The X-sample distribution F is a symmetric, k-modal distribution with support (a, b), −∞ ≤ a < b ≤ ∞, and modes R1 , . . . , Rk , 1 ≤ k < ∞. A2: The Y -sample distribution G is a symmetric, k ∗ -modal distribution with support (a∗ , b∗ ), −∞ ≤ a∗ < b∗ ≤ ∞, and modes R1∗ , . . . , Rk∗∗ , 1 ≤ k ∗ < ∞. B1: The quantity 2f ′ (t)F (t){1 − F (t)}/f 2 (t) + {2F (t) − 1} is positive (negative) for Ci < t < Ri (Ri < t < Ci∗ ), where Ci = (Ri + Ri−1 )/2, Ci∗ = (Ri + Ri+1 )/2, i = 1, . . . , k, with C1 = a and Ck∗ = b. B2: The quantity 2g ′ (t)G(t){1 − G(t)}/g 2 (t) + {2G(t) − 1} is positive (negative) ∗ for Ei < t < Ri∗ (Ri∗ < t < Ei∗ ), where Ei = (Ri∗ + Ri−1 )/2, Ei∗ = (Ri∗ + ∗ ∗ ∗ ∗ ∗ Ri+1 )/2, i = 1, . . . , k , with E1 = a and Ek∗ = b . For finite set sizes, the minimization of T (Dqp ) is computationally intensive and may depend strongly on the underlying distributions. Therefore, we first find a variance optimal design for large set sizes and then show that it works well for small set sizes and a large class of distributions. Let di and bj be percentiles based on q, p and the modes Ri , i = 1, . . . , k, and Rj∗ , j = 1, . . . , k∗ , i.e., di =  qF (Ri ), qF (Ri ) + 1 [qF (Ri ) + 1] if qF (Ri ) is integer otherwise (4) bj =  pG(Rj∗ ), pG(Rj∗ ) + 1 [pG(Rj∗ ) + 1] if pG(Rj∗ ) is integer otherwise, (5) and where [a] is the greatest integer less than a. Theorem 3 Suppose that Assumptions A1-A2 and B1-B2 hold. Then for large set sizes q and p, Dvar = {d1 , . . . , dk ; b1 , . . . , bk∗ } is variance-optimal. We emphasize that Theorem 3 provides locally optimal design and it does not stipulate that such an optimal design is unique. When the X− and Y −sample distributions are symmetric and unimodal, the optimal designs have simpler forms. 7 Corollary. If the X- and Y -sample distributions are symmetric and unimodal, then   q+1 p+1 Dvar = ; if q and p are both odd, 2 2   q+1 p p Dvar = ; , +1 if q is odd and p is even, 2 2 2   p+1 q q Dvar = , + 1; if q is even and p is odd 2 2 2 and Dvar = o nq q p p , + 1; , + 1 2 2 2 2 if q and p are both even. The non-uniqueness of the optimal design in the above corollary arises from the fact that there does not exist a unique median when the set size is even. Thus, in this case either one of the middle observations or both middle observations can be considered optimal. On the other hand, we feel that the optimal design that quantifies both middle observations provides better improvement for the MannWhitney-Wilcoxon test as suggested in Table 1. If the X- and Y -sample distributions are symmetric with bimodal bathtub shapes, then we can stipulate from Theorem 3 that the optimal design will quantify the two extreme observations in the X- and Y - sample distributions, since the modes are at the boundaries of the supports of the distributions. Theorem 4. Let F and G be asymmetric, unimodal distributions with supports (a, b), (a∗ , b∗ ), −∞ ≤ a < b ≤ ∞, −∞ ≤ a∗ < b∗ ≤ ∞, and modes R1 and R1∗ , a < R1 < b, a∗ < R1∗ < b∗ , respectively. Then the information-optimal design is Dvar = {k; k ∗ }, where k ≤ d1 and k ∗ ≤ b1 for right-skewed distributions, k ≥ d1 and k ≥ b1 for left-skewed distributions and d1 and b1 are defined in equations (4) and (5). Theorem 4 gives intervals for the possible values of k and k ∗ instead of specifying an exact design. However, our numerical calculations for the Mann-WhitneyWilcoxon test show that both k and k ∗ are equal to 1 (or q and p, respectively) for strongly right-skewed (or left-skewed) distributions and very close to 1 (or q and p, respectively) for moderately right-skewed (or left-skewed) distributions with relatively small set sizes (cf., e.g., Table 2). These numerical calculations show that the variance-optimality criterion does, indeed, lead to an improvement in the MannWhitney-Wilcoxon test. 4. IMPROVED MANN-WHITNEY-WILCOXON TEST In this section, we show that the data collection procedure based on the varianceoptimal design induced by the minimum variance criterion provides an improvement over the standard ranked set sample Mann-Whitney-Wilcoxon test. Theorems 3 and 4 indicate that two single order statistics from the X and Y distributions will maximize the information extracted from a cycle given the rank vector for a set of observations from a unimodal distribution. (Most practical applications involve data from the unimodal distributions.) Thus we restrict our search for improvement of the Mann-Whitney-Wilcoxon test to unimodal distributions 8 and consider the class of designs that require only a single order statistic in each cycle from each of the X and Y distributions. Letting D1 denote this class we point out that D1 contains pq different designs, one for each possible pair of single order statistics from each of the X and Y distributions. Theorem 5. Assume ǫ = 1/2. Then for a design D11,{di ;bj } = {di ; bj }, the Pitman ∗ = {di ; bj } with respect to the standard asymptotic relative efficiency of D11,{d i ;bj } ranked set sample is R 2{ f(di ) (y)f(bj ) (y)dy}2 σ02 (Dqp ) ∗ R ARE(D11,{di ;bj } ; DSRSS ) = , (6) {pτ1 (D11,{di ,bj } ) + qτ3 (D11,{di ,bj } )}pq{ f 2 (y)dy}2 where τ1 (D11,{di ;br } ) = bX r −1 r −1 bX j=0 u=0 q
p
p
q−1 di −1 j (j + u + di ) and τ3 (D11,{di ,br } ) = q X q X j=di u=di p p−1 br −1
q j) (j + u + br ) u
2p+q j+u+di
q
u
2p+q j+u+br −  r −1 bX
p
2  j
p+q  (j + di ) j+d i −  q X
q
2  j
p+q  . (j + br ) j+b   j=0 j=di q p q−1 di −1 p−1 br −1 r Even though Theorem 5 simplifies equation (2), it is still not feasible to obtain an analytic maximization. Thus, we numerically maximize equation (6) for several distributions, and set sizes p and q. The distributions include the normal, logistic, double exponential, Cauchy, exponential, log normal, and the gamma distribution with parameters 10 and 1. The set sizes p, q, and the ratio of the sample sizes ǫ are taken to be p, q = 1, . . . , 5 and ǫ = 1/2. Tables 1 and 2 present designs that improve the efficiency of the Mann-Whitney-Wilcoxon test for these combinations of distributions and set sizes. Table 1: Variance optimal (Dvar ) and Pitman improved (Dimp ) designs for unimodal symmetric distributions, m = n, and various set sizes. The Pitman efficiencies are given for the normal distribution. Model Symmetric and Unimodal Dist. (Normal, Logistic, Cauchy, Double Exp.) q 2 2 2 2 3 3 3 4 4 5 p 2 3 4 5 3 4 5 4 5 5 Dimp {1, 2; 1, 2} {1, 2; 2} {1; 2} or {2; 3} none {2; 2} {2; 2, 3} {2; 3} {2, 3; 2, 3} {2, 3; 3} {3; 3} Dvar {1, 2; 1, 2} {1, 2; 2} {1, 2; 2, 3} {1, 2; 3} {2; 2} {2; 2, 3} {2; 3} {2, 3; 2, 3} {2, 3; 3} {3; 3} ARE(Dimp , DSRSS ) 1 1.003 1.089 1 1.119 1.118 1.117 1.117 1.124 1.166 Since the variance-optimal designs contain two order statistics in each cycle when the set sizes are even, we also searched for the improvement in the class D2 9 Table 2: The Pitman improved designs (Dimp ) and upper limits (Dup = {d1 ; b1 }) for the variance-optimal designs (Dvar = {k1 ; k1∗ }) for selected asymmetric distributions, m = n, and various set sizes. The variance-optimal design is Dvar = {k1 ; k1∗ } such that k1 ≤ d1 and k1∗ ≤ b1 . Model Log Normal Gamma(10,1) Exp q 2 2 2 2 3 3 3 4 4 5 2 2 2 2 3 3 3 4 4 5 11 2 2 2 2 3 3 3 4 4 5 p 2 3 4 5 3 4 5 4 5 5 2 3 4 5 3 4 5 4 5 5 11 2 3 4 5 3 4 5 4 5 5 Dimp {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {2; 2} {3; 3} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} 10 Dup {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 2} {1; 2} {1; 3} {2; 2} {2; 2} {2; 3} {2, 2} {2; 3} {3; 3} {5, 5} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} {1; 1} ARE(Dimp , DSRSS ) 2.462 3.067 3.582 4.035 3.661 4.158 4.602 4.613 5.018 5.378 1.240 1.265 1.264 1.253 1.292 1.291 1.283 1.286 1.275 1.300 3.209 2.667 3.537 4.402 5.274 4.500 5.436 6.389 4.613 7.367 8.333 that contains two different order statistics in each of the X- and Y -samples. Our search algorithm first started to look for an improvement in the class D1 . After finding an optimal design with respect to the asymptotic Pitman efficiency in the class D1 , it searched for a further improvement in the class D2 . If an improved design is found in D2 , then the optimal (improved) design in D1 is updated and the search is terminated at this point. Otherwise, the optimal design in D1 is taken to be the improved design. Numerical calculations for the normal distribution showed that there was no additional improvement in the asymptotic Pitman efficiency from consideration of designs in the class D3 that contain three different order statistics in each of the X- and Y -samples. We believe that this result will also hold for other unimodal distributions and classes Dk , k = 3, . . . , a, where a is either q or p. There are several important observations from Tables 1 and 2. We first note that the result induced by the variance optimality criterion in the corollary is consistent with the designs that improve the Pitman asymptotic relative efficiency of the Mann-Whitney-Wilcoxon test for both symmetric and asymmetric distributions when the set sizes q and p are equal. For example, when the underlying distributions are symmetric and unimodal, both optimal and improved designs quantify the middle observation from each of the X and Y distributions for odd set sizes and the two middle observations for even set sizes. Similar results hold for asymmetric distributions. In this case, both optimal and improved designs quantify the smallest (largest) observation for strongly right-skewed (left-skewed) unimodal distributions. When the underlying distributions are symmetric, the designs that improve the Pitman efficiency of the Mann-Whitney-Wilcoxon test are very sensitive to the set sizes for the X and Y samples. For a given sum q + p, the highest efficiency is achieved when q = p and the improved design is consistent with the optimal design induced by the information criteria. On the other hand, if the set sizes are significantly different, there is a discrepancy between the Pitman improved designs and information optimal designs. For example, when q = 2 and p = 5, there is no design that improves the Pitman efficiency of the standard ranked set sample design of Bohn & Wolfe (1992), which indicates that there is no improvement over the standard ranked set sample procedures. When q = 2 and p = 4, the Pitman improved design is Dimp = {1; 2} or Dimp = {2; 3}, while the information optimal design is not unique and could be any of the three designs {1; 2}, {2; 3} or {1, 2; 2, 3}. Even though the optimal designs for skewed distributions are not affected by different set sizes, the efficiency is higher when the set sizes are equal. For example, for the log normal distribution, when q = 2 and p = 4 the efficiency is 3.582 but when q = 3 and p = 3 the efficiency increases to 3.661. The same result can be observed for other skewed distributions as well. Therefore, based on these observations we recommend using set sizes as equal as possible for the X and Y samples. 5. DISTRIBUTIONAL PROPERTIES OF THE IMPROVED TEST In general, the exact distribution of the standard ranked set sample MannWhitney-Wilcoxon test is not easily available because the various sequences of order statistics are not equally likely; cf., e.g., Bohn & Wolfe (1992) for a detailed discussion. In many of the improved designs, there is only one order statistic in each one of the X- and Y -samples. This makes it possible to provide an algorithm to calculate the exact null probability distribution of the test statistic when the set sizes p and q are equal. 11 Let UDimp (q; p) be the Mann-Whitney-Wilcoxon rank sum statistic based on the improved design Dimp = {d1 ; b1 }. Then we have UDimp (q, p) = m n X X i=1 j=1 δ(X(d1 )i − Y(b1 )j ). If q = p and Dopt = {d1 ; d1 }, the null distribution of UDimp (q, p) is the same as the null distribution of the simple random sample Mann-Whitney-Wilcoxon rank sum statistic based on n X’s and m Y ’s, which is available in standard nonparametric text books (cf., e.g., Hollander & Wolfe 1998). On the other hand, if the set sizes q and p are not equal, then the null distribution of UDopt (q, p) is not easily accessible. Difficulty arises from the fact that even though the number of order statistics is reduced to one in each sample, the sequences of order statistics are still not equally likely when q 6= p. Thus, UDimp (q, p), p 6= q, does not permit a simple algorithm to yield its null probabilities. APPENDIX Proof of Theorem 1. The proof follows from Theorem 3.4.13 of Randles & Wolfe (1991). We need to determine ξ1,0 (Dts ) and ξ0,1 (Dts ). First consider ξ1,0 (Dts ). From the kernel function wDts (X 1 , Y 1 ) we have EwDts (X 1 , Y 1 )wDts (X 1 , Y 2 ) − γ 2 (Dts ) ξ1,0 (Dts ) = s t X X   P {X(di )1 < min(Y(br )1 , Y(br )2 )} − P 2 (X(di )1 < Y(br )1 ) = i=1 r=1 t XX X + i=1 1≤r6=ℓ≤s  P {X(di )1 < min(Y(br )1 , Y(br )2 )}  −P (X(di )1 < Y(br )1 )P (X(di )1 < Y(bl )1 ) τ1 (Dts ) + τ2 (Dts ). = It then follows that "Z Z 2 # s t X X 2 {1 − Fbr (y)} dF(di ) (y) − {1 − Fbr (y)}dF(di ) (y) τ1 (Dts ) = i=1 r=1 = s t X X i=1 r=1 × Z  − =  bX r −1 r −1 bX  j=0 k=0 q  q−1 di − 1    p p j k F j+k+di −1 (y){1 − F (y)}2p−j−k+q−di f (y)dy bX r −1 j=0 s t X X i=1 r=1 2    Z p q−1  F j+di −1 (y){1 − F (y)}p−j+q−di f (y)dy   q di − 1 j   bX r −1 r −1 bX  j=0 k=0 q−1 p di −1 j
2p+q j+k+di (j + q

p
k k + di ) −
p
)2  q−1 q j=0 ,
di −1 j p+q j+di (j + di ) ( Pbr −1 which is the expression in Theorem 1. The expression τ2 (Dts ) follows from a similar calculation. 12 We now consider ξ0,1 (Dts ). Again from the kernel function wDts (X 1 , Y 1 ) we have ξ0,1 (Dts ) = = EvDts (X 1 , Y 1 )wDts (X 2 , Y 1 ) − γ 2 (Dts ) s t X X  i=1 r=1 + =  P {max(X(di )1 , X(di )2 ) < Y(br )1 } − P 2 (X(di )1 < Y(br )1 ) XX 1≤i6=j≤t s X  P {max(X(di ) , X(dj )1 ) < Y(br )1 } r=1  −P (X(di )1 < Y(br )1 )P (X(dj )1 < Y(br )1 ) τ3 (Dts ) + τ4 (Dts ). The remaining part of the proof is completed by evaluating the corresponding probabilities as done for ξ1,0 (Dts ). The details are omitted. ∗ is an arbitrary design and contains SRSS, Proof of Theorem 2. Since the design Dts ∗ for the proof of the theorem it is sufficient to derive the Pitman efficacy of Ū (Dts ). ∗ (·, ·) we get From the kernel function wDts ∗ ∗ µ′Dts ∗ (0) = t s s Z t X X f(di ) (y)f(bj ) (y)dy. (7) i=1 j=1 ∗ , the Theorem 1 yields Under the null hypothesis, again for the design Dts ∗ )= σ0 (D(ts) t∗ s∗ 2 ξ1,0 (Dts ) s∗ t∗ 2 ξ0,1 (Dts ) + . ǫ 1−ǫ (8) The proof is completed by putting equations (7) and (8) in equation (1). Proof of Theorem 3. For the proof of that theorem, we need to minimize T (D) with respect to D. For large set sizes q and p, T (D) can be written as T ∗ (λ) = q p ǫ X λ∗j (1 − λ∗j ) 1 − ǫ X λi (1 − λi ) + , ǫ1 i=1 f 2 (ηλi ) 1 − ǫ1 j=1 g 2 (ηλ∗∗ ) (9) j where λ = {λ1 , . . . , λq ; λ∗1 , . . . , λ∗p }, 0 < λi = limq→∞ di /q < 1, i = 1, . . . , q; 0 < λ∗j = limp→∞ bj /p < 1, j = 1, . . . , p; ηλi = F −1 (λi ), and ηλ∗∗ = G−1 (λ∗j ). The j proof is completed by applying Theorem 1 of Öztürk & Wolfe (1998a) to each term in equation (9). Proof of Theorem 4. The proof follows from a consideration similar to that of the proof of Theorem 2 in Öztürk & Wolfe (1998a). ACKNOWLEDGMENTS The authors thank the Editor and anonymous referees for their helpful comments which improved the presentation. The work of Douglas A. Wolfe was supported in part by National Science Foundation Grant Number DMS-9802358. 13 REFERENCES Bhoj, D. S. (1997). New parametric ranked set sampling. J. Appl. Statist. Sci., 6, 275–289. Bohn, L. L. (1998). A ranked-set sample signed-rank statistic. J. Nonpar. Statist., 9, 295–306. Bohn, L. L., & Wolfe, D. A. (1992). Nonparametric two-sample procedures for ranked-set samples data. J. Amer. Statist. Assoc., 87, 552–561. Bohn, L. L., & Wolfe, D. A. (1994). The effect of imperfect judgment rankings on properties of procedures based on the ranked-set samples analog of the MannWhitney-Wilcoxon statistic. J. Amer. Statist. Assoc., 89, 168–176. Dell, T. R., & Clutter, J. L. (1972). Ranked-set sampling theory with order statistics background. Biometrics, 28, 545–555. Halls, L. K., & Dell, T. R. (1966). 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On unbiased estimates of the population mean based on the sample stratified by means of ordering. Ann. Inst. Statist. Math., 20, 1–31. 14 Received 26 March 1999 Accepted 11 June 1999 Ömer Öztürk Department of Statistics The Ohio State University Marion, Ohio 43302 USA e-mail: [email protected] Douglas A. Wolfe Department of Statistics The Ohio State University Columbus, Ohio U.S.A. 43210 e-mail: [email protected] 15