More Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order Restriction

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1 Sankhyā : The Indian Journal of Statistics 2007, Volume 69, Part 4, pp c 2007, Indian Statistical Institute More Powerful Tests for Homogeneity of Multivariate Normal Mean Vectors under an Order Restriction Shoichi Sasabuchi Kyushu University, Japan Abstract Consider the problem of testing the homogeneity of several p-variate normal mean vectors under an order restriction. This is a multivariate extension of Bartholomew s (Biometrika, 1959) problem. When the covariance matrices are known, this problem has been studied to some extent, for example, by Sasabuchi, Inutsuka and Kulatunga (Biometrika, 1983), Sasabuchi, Miura and Oda (JSCS, 2003) and some others. We are interested in the case when the covariance matrices are common but unknown. In this case, Sasabuchi, Tanaka and Tsukamoto (Ann. Statist., 2003) proposed a test statistic and studied its upper tail probability under the null hypothesis. In the present paper, we provide some tests, which are more powerful than the above test. We derive some theorems about their null distributions and powers. AMS (2000) subject classification. Primary 62F30; secondary 62F03, 62H15. Keywords and phrases. Common unknown covariance matrix, isotonic regression, multivariate isotonic regression, multivariate normal distribution, similar test. 1 Introduction Bartholomew (1959) considered the problem of testing the homogeneity of several univariate normal means against an order restricted alternative hypothesis. He derived the likelihood ratio test statistic, χ 2 k, and its null distribution under the assumption that the variances are known. Since then, an extensive literature concerning this problem has appeared, and most of it has been summarized by Barlow et al. (1972), Robertson et al. (1988), and Silvapulle and Sen (2005). It has been shown that, for the order restricted alternative hypothesis, the χ 2 k test is more powerful than the usual χ 2 test, which is the likelihood ratio test for testing homogeneity against the unrestricted alternative hypothesis.

2 Tests for homogeneity of multivariate normal means 701 We are interested in a multivariate extension of Bartholomew s (1959) problem. Consider p-variate normal distributions N p (µ i, i ), i = 1, 2,..., k. We are concerned with the problem of testing : µ 1 = µ 2 = = µ k versus H 1 : µ 1 µ 2 µ k but does not hold, where µ i µ j means that all the elements of µ j µ i are non-negative. This problem may arise in the situation, where the values of several parameters increase simultaneously. For example, if we wish to know whether both the average height and the average weight of the children of an area increase simultaneously year by year, we could apply our test to the bivariate data sets obtained by random sampling from the population. When the covariance matrices are known, this problem has been studied to some extent. Sasabuchi et al. (1983) derived the likelihood ratio test statistic, χ 2 k,p, Kulatunga and Sasabuchi (1984) studied its null distribution, and Nomakuchi and Shi (1988) proposed a new test whose null distribution was easier to calculate than that of the χ 2 k,p test. Sasabuchi et al. (1998) made some power comparisons by simulation in the bivariate case and showed that over H 1 the χ 2 k,p test is more powerful than the usual χ2 test, which is the likelihood ratio test for against the unrestricted alternative hypothesis. Sasabuchi, Miura and Oda (2003) further studied this problem in the higher dimensional cases and showed similar results. In the present paper, we assume that the covariance matrices are common but unknown. When p = 1, this case has been studied by Bartholomew (1961) and many others, and the likelihood ratio test statistic is well-known as Ē2 k. We are interested in its multivariate extension. For the multivariate case, the likelihood ratio test statistic for against the unrestricted alternative hypothesis was given by Anderson (1984, Section 8.8). But, in our problem, the alternative has an order restriction, thus we should take this restriction into consideration. To the author s knowledge, the likelihood ratio test for versus H 1 in the case, when the covariance matrices are common but unknown, has not been obtained yet. Perlman (1969) studied a multivariate one-sided testing problem with an unknown covariance matrix and derived its likelihood ratio test. It may seem that our problem may be reduced to his, but the structure of our model is different from his, and as a result we cannot apply his methods or results directly to our problem.

3 702 Shoichi Sasabuchi Sasabuchi, Tanaka and Tsukamoto (2003) proposed a test statistic, T 2, by replacing the unknown covariance matrix in χ 2 k,p by its estimator. They studied the upper tail probability of T 2 under. In the present paper, we provide some tests, which are more powerful than T 2 test. We derive some theorems about their null distributions and powers. In Section 2, we review the problem and results given by Sasabuchi, Tanaka and Tsukamoto (2003). We define a set of new tests, T m tests, and present some theorems about them in Section 3. In Section 4, some geometrical interpretations of the test statistics are given. In Section 5, we consider some omnibus tests combining T m tests and study their properties. In Section 6, some discussions are given. The proofs of the theorems are given in Section 7. 2 The Problem and Preliminary Results Suppose that X i1,..., X ini are random samples from a p-variate normal distribution N p (µ i, ), i = 1, 2,..., k. We assume that is unknown and N N k > p + k. Throughout this paper, all vectors are column vectors as a rule. As usual, for any vector x and matrix A, x and A denote their transposes, respectively. Consider the problem of testing : µ 1 = µ 2 = = µ k versus H 1 : µ 1 µ 2 µ k but does not hold. The restriction imposed on H 1 means that all the components of the p- dimensional mean vector µ i increase simultaneously as i increases. In order to discuss the order restricted problem, we give some definitions. Definition 2.1 (Barlow et al., 1972). Given real numbers x 1,..., x k and positive numbers w 1,..., w k, a k-dimensional real row vector (ˆθ 1,..., ˆθ k ) is said to be the isotonic regression (IR) of x 1,..., x k with weights w 1,..., w k if ˆθ1 ˆθ 2 ˆθ k, and the row vector (ˆθ 1,..., ˆθ k ) satisfies min θ 1 θ 2 θ k (x i θ i ) 2 w i = (x i ˆθ i ) 2 w i.

4 Tests for homogeneity of multivariate normal means 703 The isotonic regression can be computed easily by the well-known method, Pool-Adjacent-Violators Algorithm; see Barlow et al. (1972). Definition 2.2 (Sasabuchi et al., 1983). Given p-dimensional real vectors x 1,..., x k and p p positive definite matrices Λ 1,..., Λ k, a p k real matrix (ˆθ 1,..., ˆθ k ) is said to be the multivariate isotonic regression (MIR) (in fact, p-variate isotonic regression) of x 1,..., x k with weights Λ 1 1,..., Λ 1 k if ˆθ 1 ˆθ 2 ˆθ k, and the matrix (ˆθ 1,..., ˆθ k ) satisfies min θ 1 θ 2 θ k (x i θ i ) Λ 1 i (x i θ i ) = (x i ˆθ i ) Λ 1 i (x i ˆθ i ), where θ i θ j means that all the elements of θ j θ i are non-negative. Sasabuchi et al. (1983) proposed an iterative algorithm for the computation of MIR and studied its convergence in the case when p = 2. This algorithm was extended to the general multivariate case by Sasabuchi et al. (1992). Let us put X i = 1 N i X ij, i = 1,..., k, N i X = S = j=1 ( ) 1 N i N i N i X i, and (X ij X i )(X ij X i ). j=1 Note that S has the Wishart distribution W p (N k; ), where N = N N k, and is statistically independent of X 1,..., X k. Further, let ( µ 1,..., µ k ) denote the MIR of X 1,..., X k with weights N 1 S 1,..., N k S 1. Sasabuchi, Tanaka and Tsukamoto (2003) proposed the following test and studied its null distribution. T 2 = N i ( µ i X) S 1 ( µ i X) c reject, where c is a positive constant depending on the significance level. In the present paper, we call this test the T 2 -test.

5 704 Shoichi Sasabuchi In order to discuss the upper tail probability of T 2 under, Sasabuchi, Tanaka and Tsukamoto (2003) introduced the following statistic. T = N i (X i X) S 1 (X i X) 1 s 11 N i (X i1 µ i1 ) 2, where X i1 is the first component of X i, i = 1, 2,..., k, s 11 is the (1, 1)-th element of S, and ( µ 11,..., µ k1 ) is the IR (univariate isotonic regression) of X 11,..., X k1 with weights N 1,..., N k. Sasabuchi, Tanaka and Tsukamoto (2003) derived the following three theorems. Theorem 2.1. Under, the distribution of T 2 is independent of µ 0, where µ 0 is the common value of µ 1,..., µ k. According to this theorem, we can assume that µ 1 = = µ k = 0 in computing the upper tail probability of T 2 under. But it still depends on the unknown. Theorem 2.2. Under, the distribution of T is independent of µ 0 and, where µ 0 is the common value of µ 1,..., µ k. According to this theorem, we can assume that µ 1 = = µ k = 0 and = I p, the identity matrix, in computing the upper tail probability of T under. Theorem 2.3. For any real number c, Pr (T 2 c) = Pr (T 2 c) = Pr (T c). 0, 0,I p Here Pr denotes the probability measure corresponding to the parameters µ = (µ 1,..., µ k ) and, the remum with respect to denotes the remum over µ 1,..., µ k with µ 1 = = µ k, and the remum with respect to denotes the remum over all the p p positive definite real matrices.

6 Tests for homogeneity of multivariate normal means 705 By Theorem 2.3, in order to get the upper α point of T 2 under, we only need to obtain that of T when µ = 0 and = I p. We do not know the exact distribution of T, but we can easily get the approximate value of Pr (T c) using Monte Carlo simulation by generating standard 0,I p normal random numbers, because T is easy to calculate, and we need the distribution under µ = 0 and = I p. The upper α points of T estimated by Monte Carlo simulation were given in Table 1 in Sasabuchi, Tanaka and Tsukamoto (2003, Section 6). 3 A Set of Tests which are More Powerful than the T 2 -test In the present paper, we define p statistics: T m, m = 1, 2,..., p, by T m = N i (X i X) S 1 (X i X) 1 s mm N i (X im µ im ) 2, m = 1, 2,..., p, where, for each m = 1, 2,..., p, X im is the m-th component of X i, i = 1, 2,..., k, s mm is the (m, m)th element of S, and ( µ 1m,..., µ km ) is the IR (univariate isotonic regression) of X 1m,..., X km with weights N 1,..., N k. Note that T 1 coincides with T. For each T m, m = 1, 2,..., p, we can prove the following theorem. Theorem 3.1. (i) T m T 2. (ii) Under, the distribution of T m is independent of µ 0 and, where µ 0 is the common value of µ 1,..., µ k. (iii) For any real number c, Pr (T 2 c) = Pr (T 2 c) = Pr (T m c). 0, 0,I p (iv) For any real number c, Pr 0,I p (T m c) = Pr 0,I p (T c).

7 706 Shoichi Sasabuchi According to (ii), we can assume that µ 1 = = µ k = 0 and = I p, the identity matrix, in computing the upper tail probability of T m under. Now, for each m = 1, 2,..., p, we consider the following procedure for testing versus H 1. If T m c m, reject, where c m is a positive constant depending on the significance level, m = 1, 2,..., p. We call this test the T m -test, m = 1, 2,..., p. Directly from Theorem 3.1, we can get the following theorem for each T m, m = 1, 2,..., p. Theorem 3.2. (i) For any real number c, and for any µ and, Pr (T m c) Pr (T 2 c). (ii) The T m -test is a similar test for testing. (iii) If we put c m = c, then the T m -test and the T 2 -test have the same significance levels. Further, we can prove a stronger result as follows. Theorem 3.3. For any positive number c, and for any µ and, Pr (T m c) > Pr (T 2 c), which implies that the T m -test is uniformly and strictly more powerful than the T 2 -test if we put c m = c > 0. Thus, we have got p tests: the T m -test, m = 1, 2,..., p, all of which are uniformly and strictly more powerful than the T 2 -test. By (iii) of Theorem 3.2, the upper α points of T m are just the same as those of T 2, and thus we can use Table 1 given in Section 6 of Sasabuchi, Tanaka and Tsukamoto (2003).

8 Tests for homogeneity of multivariate normal means Geometrical Interpretation In this section, we present some geometrical interpretations of the statistics T m, m = 1, 2,..., p. Let R p and R pk denote the p-dimensional and pk-dimensional real Euclidean spaces, respectively. (Note that all vectors are column vectors as mentioned in Section 2.) Let x 1,..., x k, y 1,..., y k be p-dimensional real vectors and put x = x 1. x k, y = y 1. y k. Define an inner product, S and a norm S in R pk by x, y S = N i x is 1 y i and x S = x, x 1/2 S, respectively. Let C be a closed convex cone in R pk. For x R pk, the orthogonal projection of x onto C with respect to, S, denoted by π S (x; C), is defined by the point which minimizes x z S under the restriction that z C. Put X = X 1. X k. Define two closed convex cones C 0 and C 1 in R pk by µ 1 µ 2 C 0 = µ = µ. 1 = µ 2 = = µ k, µ i R p, i = 1,..., k and µ k µ 1 µ 2 C 1 = µ = µ. 1 µ 2 µ k, µ i R p, i = 1,..., k. Then, we can write µ k T 2 = π S (X; C 1 ) π S (X; C 0 ) 2 S = X π S (X; C 0 ) 2 S X π S (X; C 1 ) 2 S.

9 708 Shoichi Sasabuchi See Sasabuchi, Tanaka and Tsukamoto (2003) for more detailed explanations. Now, we define p closed convex cones C (m), m = 1, 2,..., p, in R pk by µ 1 µ 2 C (m) = µ = µ. 1m µ 2m µ km, µ i R p, i = 1,..., k, µ k where µ im denotes the m-th component of µ i, i = 1,..., k. Then, in a manner similar to that in Sasabuchi, Tanaka and Tsukamoto (2003), we can show that T m = X π S (X; C 0 ) 2 S X π S (X; C (m) ) 2 S, m = 1, 2,..., p. Note that, since C 0 C 1 C (m), m = 1, 2,..., p, we have X π S (X; C 0 ) 2 S X π S (X; C 1 ) 2 S X π S (X; C (m) ) 2 S, and thus T m T 2 0, m = 1, 2,..., p. m = 1, 2,..., p, For the discussion given in the rest of this section, readers are referred to, for example, Silvapulle and Sen (2005, Section 3.12). For any closed convex cone C in R pk, let C o denote the negative dual cone or polar cone of C, that is, C o = {x R pk x, y S 0 for all y C}. Then, it is important to note that π S (x; C) = 0 if and only if x C o. Since C (m) is a polyhedral cone defined by (k 1) independent linear inequality constraints, C(m) o is also a polyhedral cone generated by (k 1) linearly independent vectors in R pk. Similarly, since C 1 is a polyhedral cone defined by p(k 1) independent linear inequality constraints, C1 o is also a polyhedral cone generated by p(k 1) linearly independent vectors in R pk. Then, we have that C1 o = Co (1) + + Co (p), the direct sum of Co (1),..., Co (p), since C 1 = C (1) C (p). The relative interior of C1 o, denoted by ri(co 1 ), is defined as the interior of C1 o with respect to the relative topology induced in the p(k 1)-dimensional linear subspace spanned by C1 o.

10 Tests for homogeneity of multivariate normal means 709 Now pose that a point x lies in ri(c1 o). Then, π S(x; C 1 ) = 0 because x C1 o, and thus x π S(x; C 1 ) 2 S = x 2 S. While, for m = 1, 2,..., p, π S (x; C (m) ) 0 because x / C(m) o, and thus x π S(x; C (m) ) 2 S = x 2 S π S (x; C (m) ) 2 S < x 2 S. Therefore, if X lies in ri(c1 o), we have T m > T 2, m = 1, 2,..., p, and hence, min T m > T 2. We will use these facts in the proofs of the theorems 1 m p in Section 7. 5 Omnibus Tests Combining the T m -Tests When is known, the likelihood ratio test statistic, χ 2 k,p, for versus H 1 was given by Sasabuchi et al. (1983). Sasabuchi, Tanaka and Tsukamoto (2003) proposed the test statistic T 2 by replacing in χ 2 k,p by its estimator. In this sense, using T 2 is considered to be intuitively reasonable when is unknown. But, by Theorem 3.3, the T 2 -test is less powerful than the T m - test, and thus not admissible. More precisely, the T 2 -test is not α-admissible in the sense of Lehmann (1986). Besides, the calculation of the statistic T m is much easier than that of T 2. We need to compute MIR by using some complicated computer algorithm when we calculate T 2. While, we only need to compute IR by using Pool- Adjacent-Violators Algorithm, a very simple method, when we calculate T m. We now need to address the question: which test should we choose among the T m -tests, m = 1, 2,..., p. Here, we observe some properties of these statistics. Recall that T m = X π S (X; C 0 ) 2 S X π S (X; C (m) ) 2 S, m = 1, 2,..., p, as written in Section 4. The second term of the right-hand side is the distance between X and C (m), and hence the value of this term tends to become small when the true value of µ belongs to C (m), that is, when µ 1m µ 2m µ km. Therefore, T m is a test statistic that is sensitive to the simple order restriction among µ 1m, µ 2m,..., µ km, the m-th components of µ 1, µ 2,..., µ k. Thus, we should not choose only one statistic from T m, m = 1, 2,..., p. We should use some omnibus tests combining the T m -tests. First, consider two reasonable candidates of the omnibus test statistics defined by T MAX = max 1 m p T m

11 710 Shoichi Sasabuchi and T AV E = T Average = 1 p p T m. m=1 By Theorem 3.1, T MAX T 2 and T AV E T 2, and hence, both the T MAX -test and the T AV E -test are expected to be more powerful than the T 2 - test if we can use the same critical point as that of T 2. But, even under, the distributions of T MAX and T AV E depend on since the joint distribution of (T 1, T 2,..., T p ) depends on, although the marginal distributions of T m s, m = 1, 2,..., p, do not. Thus we need to find some upper bounds of the upper tail probabilities of these statistics under. We can prove the following theorem. Theorem 5.1. For any real number c, Pr (T c) 0,I p Pr (T c) 0,I p Pr (T MAX c) p Pr (T c), 0,I p Pr (T AV E c) p Pr (T c). 0,I p However, these upper bounds are too conservative. For example, when p = 5, if we wish to test versus H 1 with significance level 0.05, then, we have to use the upper 0.01 point of T. Thus, unfortunately, we cannot use the same upper α points as those of T 2. Conservativeness yields some power reduction. We need a sharper upper bound but we have not got it yet. Now, we consider another omnibus test statistic defined by T MIN = min T m. 1 m p The null distribution of T MIN also depends on. But we can show the following theorem. Theorem 5.2. For any real number c, Pr (T MIN c) = Pr (T c). 0,I p Therefore, we can use the same upper α points as those of T 2. Further, we can prove the following theorem.

12 Tests for homogeneity of multivariate normal means 711 Theorem 5.3. For any positive number c, and for any µ and, Pr (T MIN c) > Pr (T 2 c), which implies that the T MIN -test is uniformly and strictly more powerful than the T 2 -test. Furthermore, the calculation of the statistic T MIN is much easier than that of T 2. We only need to compute IR instead of MIR. At present, the T MIN -test can be recommended because it is not only much easier to calculate than the T 2 -test but also uniformly and strictly more powerful than the T 2 -test. 6 Discussions 6.1. On the derivation of the likelihood ratio test. Perlman (1969) studied the following problem. Let Y 1,..., Y N be a random sample from a p-variate normal distribution N p (η, ), where is unknown and N > p. N Y i and S = N (Y i Y )(Y i Y ). Consider the problem Let Y = 1 N of testing H : η = 0 versus K : η 0. Let X = N 1 2 Y and µ = N 1 2 η. As given in Perlman (1969), the likelihood ratio test statistic can be expressed in quite a simple form by using the minimum value of (X µ) S 1 (X µ) under the restriction that µ 0. On the other hand, as stated in Section 7 of Sasabuchi, Tanaka and Tsukamoto (2003), in order to get the likelihood ratio test for our problem in the present paper, we need the minimum value of the determinant I p + S 1/2 N i (X i µ i )(X i µ i ) S 1/2 under the restriction that µ 1 µ 2 µ k. (Note that this S is the one defined in Section 2.) The difficulty in deriving the likelihood ratio test arises due to the difficulty in solving this minimization problem explicitly. The difference between the difficulties of the two problems is caused by the structural difference of the models. Perlman (1969) considered one multivariate population, while we consider several multivariate populations. One might expect that our problem could be transformed into Perlman s

13 712 Shoichi Sasabuchi (1969) problem by some linear transformations on the variables. But, if we had transformed the variables, the covariance structure of the resultant variables would not have satisfied the setup assumed in Perlman (1969). Thus, we cannot apply his technique directly to our problem On the implications of the proposed test statistics. Note that T 2 is a statistic for testing versus H 1, while T m is a statistic sensitive to the order restriction only among the m-th components (m = 1, 2,..., p). The statistic T 2 was defined as an analogy to the likelihood ratio test statistic in the case when is known. But this is not the case with T m. The test T m was constructed by extending the critical region of the T 2 -test, considering the fact that the T 2 -test is not a similar test for. We explain this as follows. First, note that, for any positive number c, {(X, S) T 2 c} {(X, S) T MIN c} {(X, S) T m c}, because T 2 T MIN T m, m = 1, 2,..., p. Pr (T 2 c) depends on under, and thus the T 2 -test is conservative for any given, since is unknown. While Pr (T m c) does not depend on (and µ) under, hence the T m -test is a similar and exact test for. The region {(X, S) T m c} is constructed by extending the region {(X, S) T 2 c} so that its probability measure does not depend on under. We have p directions of such extension, and thus we get p statistics: T m, m = 1, 2,..., p. For each m = 1, 2,..., p, the T m -test is indeed more powerful than the T 2 -test and the T MIN -test. But {(X, S) T m c} is too wide a region and pays attention only to the m-th components, ignoring the other (p 1) components. Thus, the T m -test should not be recommended as the test for versus H 1. On the other hand, the T MIN -test pays attention to all the components and is still more powerful than the T 2 -test. 7 Proofs of the Theorems Proof of Theorem 3.1. First, note that T 1 = T. Hence, for m = 1, (ii) and (iii) coincide with Theorems 2.2 and 2.3, respectively. According to the proofs of Lemma 5.3 and Theorem 3.2 of Sasabuchi, Tanaka and Tsukamoto (2003), we have T T 2, and hence (i) also holds for m = 1.

14 Tests for homogeneity of multivariate normal means 713 By using similar methods, we can see that (i), (ii) and (iii) hold also for m = 2,..., p. Finally, (iv) follows directly from (iii) and Theorem 2.3. Proof of Theorem 3.3. We will prove the theorem for m = 1. By using similar methods, we can see that the theorem holds also for m = 2,..., p. Let S and any positive number c be fixed. First, recall that T 1 T 2 0 in general. It is also important to note that the port of the probability distribution of X = (X 1, X 2,..., X k) is R pk for any µ and. Let A denote the set of all the sample points of X, which satisfy T 1 > T 2. Note that A is not empty because of the fact shown in the last part of Section 4. Both T 1 and T 2 are continuous functions of X, hence A is an open set, and thus the Lebesgue measure of A in R pk is positive. For any ε > 0, let A(ε) denote the set of all the sample points of X which satisfy T 1 > ε > T 2. Then, there exists a positive number ε 0 for which the Lebesgue measure of A(ε 0 ) in R pk is also positive. For any δ > 0, define the set δa(ε 0 ) in R pk by δa(ε 0 ) = {δx x A(ε 0 )}. It is trivial to show that, for any δ > 0, the Lebesgue measure of δa(ε 0 ) in R pk is also positive. Further, we can easily see that δa(ε 0 ) = A(δ 2 ε 0 ) from the definitions of T 1 and T 2. Hence, if we put δ 0 = (c/ε 0 ) 1/2, then A(c) = A(δ 2 0 ε 0) = δ 0 A(ε 0 ), and thus the Lebesgue measure of the set of all the sample points of X which satisfy T 1 >c>t 2 in R pk, is also positive. Since X = (X 1, X 2,..., X k) is statistically independent of S, and the port of the probability distribution of X is R pk, we have Pr (T 1 > c > T 2 ) > 0, for any µ and. Therefore, we finally get Pr (T 1 c) = Pr (T 1 c, T 2 c) + Pr (T 1 c, T 2 < c) = Pr (T 2 c) + Pr (T 1 c > T 2 ) Pr (T 2 c) + Pr (T 1 > c > T 2 ) > Pr (T 2 c) for any µ and. This completes the proof.

15 714 Shoichi Sasabuchi Proof of Theorem 5.1. Note that T 2 T MAX. Hence, we have Pr (T 2 c) = = p = m=1 m=1 Pr (T MAX c) Pr ( max T m c) 1 m p Pr (T m c for some m, 1 m p) p Pr (T m c) m=1 p Pr (T c) 0,I p = p Pr 0,I p (T c) Pr (T m c) by using Theorem 3.1. Similarly, noting that T 2 T AV E, we also have Pr (T 2 c) Pr (T AV E c) p Pr (T c). 0,I p On the other hand, by Theorem 2.3, we have Pr (T 2 c) = Pr (T c), 0,I p and thus we complete the proof. Proof of Theorem 5.2. we have Note that T 2 T MIN T 1 = T. Hence, Pr (T 2 c) = Pr 0,I p (T c) Pr (T MIN c) Pr (T c) by using Theorem 2.2. On the other hand, by Theorem 2.3, we have Pr (T 2 c) = Pr 0,I p (T c),

16 Tests for homogeneity of multivariate normal means 715 and thus the proof is complete. Proof of Theorem 5.3. Let B denote the set of all the sample points of X which satisfy T MIN > T 2. Note that B is not empty because of the fact shown in the last part of Section 4. Then, we can apply exactly the same technique as that in the proof of Theorem 3.3 to complete the proof. Acknowledgements. The author is deeply grateful to the referees for their valuable comments and suggestions. References Anderson, T.W. (1984). An Introduction to Multivariate Statistical Analysis, 2nd edition, Wiley, New York. Barlow, R.E., Bartholomew, D.J., Bremner, J.M. and Brunk, H.D. (1972). Statistical Inference under Order Restrictions: The Theory and Application of Isotonic Regression. Wiley, New York. Bartholomew, D.J. (1959). A test of homogeneity for ordered alternatives, Biometrika, 46, Bartholomew, D.J. (1961). Ordered tests in the analysis of variance, Biometrika, 48, Kulatunga, D.D.S. and Sasabuchi, S. (1984). A test of homogeneity of mean vectors against multivariate isotonic alternatives, Mem. Fac. Sci. Kyushu Univ. Ser. A (Math.), 38, Lehmann, E.L. (1986). Testing Statistical Hypotheses, 2nd edition, Wiley, New York. Nomakuchi, K. and Shi, N.-Z. (1988). A test for a multiple isotonic regression problem, Biometrika, 75, Perlman, M.D. (1969). One-sided testing problems in multivariate analysis. Ann. Math. Statist., 40, Robertson, T., Wright, F.T. and Dykstra, R.L. (1988). Order Restricted Statistical Inference, Wiley, New York. Sasabuchi, S., Inutsuka, M. and Kulatunga, D.D.S. (1983). A multivariate version of isotonic regression, Biometrika, 70, Sasabuchi, S., Inutsuka, M. and Kulatunga, D.D.S. (1992). An algorithm for computing multivariate isotonic regression, Hiroshima Math. J., 22, Sasabuchi, S., Kulatunga, D.D.S. and Saito, H. (1998). Comparison of powers of some tests for testing homogeneity under order restrictions in multivariate normal means, Amer. J. Math. Management Sci., 18, Sasabuchi, S., Miura, T. and Oda, H. (2003). Estimation and test of several multivariate normal means under an order restriction when the dimension is larger than two, J. Statist. Comput. Simulation, 73, Sasabuchi, S., Tanaka, K. and Tsukamoto, T. (2003). Testing homogeneity of multivariate normal mean vectors under an order restriction when the covariance matrices are common but unknown, Ann. Statist., 31,

17 716 Shoichi Sasabuchi Silvapulle, M.J. and Sen, P.K. (2005). Constrained Statistical Inference: Inequality, Order, and Shape Restrictions, Wiley, New York. Shoichi Sasabuchi Department of Applied Information and Communication Sciences Faculty of Design Kyushu University 4-9-1, Shiobaru, Minami-Ku Fukuoka-Shi, , Japan Paper received January 2007; revised September 2007.