On the powers of tests for homogeneity of regression coefficient vectors under synchronised order restrictions

We consider a multivariate multiple linear regression model and study the problem of testing homogeneity of regression coefficient vectors under synchronised order restrictions when the covariance matrices are common but unknown. Synchronised order restrictions are the generalisation of the multivariate isotonic order restrictions. Synchronised order restricted test could be applied to a situation where the values of some parameters increase, those of some other parameters decrease, and those of the rest of the parameters have no restriction, simultaneously. For this problem, some test statistics were proposed and some inequalities among their powers were obtained in the past. This showed that the proposed test statistics may equally be good in terms of their powers. In the present paper, we mathematically prove that the strict inequalities hold among the powers of the test statistics. Thus we attain an exact comparison among the powers of the test statistics indicating more accurate and stronger results than those obtained in the past.


INTRODUCTION
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 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 them have been summarised by Barlow et al. (1972), Robertson et al. (1988), and Silvapulle and Sen (2005). Sasabuchi et al. (1983) considered the problem of testing the homogeneity of several p-variate normal mean vectors against the alternative hypothesis determined by a multivariate isotonic order restriction. This is a multivariate extension of Bartholomew's (1959) problem. When the covariance matrices are known, this problem has been studied by Sasabuchi et al. (1983;1998;2003a), Kulatunga and Sasabuchi (1984), Nomakuchi and Shi (1988) and some others.
When the covariance matrices are common but unknown, Sasabuchi et al. (2003b)  are non-negative, proposed a t are non-negative, proposed a test statistic, and studied its null distribution. Sasabuchi (2007) considered the same problem and provided some tests, which are more powerful than the test proposed in Sasabuchi et al. (2003b). Hu (2009)  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, then we could apply our test to the bivariate datasets obtained by random sampling from the population. Hu and Banerjee (2012) considered a multivariate multiple linear regression model and studied the problem of testing homogeneity of regression coefficient vectors under synchronised order restrictions when the covariance matrices are common but unknown. For this problem, Hu and Banerjee (2012) proposed some test statistics, studied their distributions, and obtained some inequalities among their powers.
Synchronised order restrictions are the generalisation of the multivariate isotonic order restrictions. Synchronised order restricted test could be applied to a situation where the values of some parameters increase, those of some other parameters decrease, and those of the rest of the parameters have no restriction, simultaneously. An example of such situation has been given in section 2.1 of Hu and Banerjee (2012). Besides, they (Hu & Banerjee, 2012) have considered general partial orderings including the simple ordering considered by Sasabuchi et al. (2003b). Furthermore, multivariate multiple linear regression model is also a generalisation of several p-variate normal distributions model. Thus the problem of Hu and Banerjee (2012) is considered to be the generalisation of that of Sasabuchi et al. (2003b) in three senses.
The main objective of this paper is to prove mathematically that the strict inequalities hold among the powers of the tests given in Hu and Banerjee (2012). In order to prove them some careful geometrical discussion is needed and they are elucidated in the text. The exact results among the powers of the test statistics obtained in this paper are stronger than those obtained in Hu and Banerjee (2012).
In the next section, we review the problem and summarise Hu and Banerjee's (2012) findings. The main theorem and its proof are given later. Proofs of some lemmas are given in the Appendix.

PRELIMINARIES
In this section, we summarise Hu and Banerjee's (2012) findings.
Throughout this paper, column vectors are considered as vectors. In the usual notation, sidered as vectors. In the usual notation,   This problem is a generalisation of that considered by Sasabuchi et al. (2003b). As stated in the introduction, the problem considered by Sasabuchi et al. (2003b) may arise in the situation where the values of several parameters increase simultaneously. While the above problem could be applied to the situation where the values of some parameters increase, those of some other parameters decrease, and those of the rest of the parameters have no restriction, simultaneously. An example of such situation was given in section 2.1 of Hu and Banerjee (2012). Besides, now we consider general partial order: Let q including the simple order considered by Sasabuchi et al. (2003b). Furthermore, multivariate multiple linear regression model is also a generalisation of several p-variate normal distributions model. Thus the above problem is considered to be the generalisation of that of Sasabuchi et al. (2003b) in three senses. Now we define the inner product, norm and projection in R p×q as follows.
Let V be a p×p positive definite symmetric matrix.
define an inner product of A and B, and a norm of A, respectively.

Let
Let be a closed convex cone in be a closed convex cone in R p×q . Then, for any ).
Hu and Banerjee (2012) proposed the following test statistic: The notation y one), thus the above

Here
Here , P denotes the probability measure corresponding to denotes the probability measure corresponding to the parameters q p X q n is any p×p positive definite symmetric matrix, and is any positive definite symmetric matrix, and p I is the is the p×p identity matrix. Note that we have changed the notation given in Hu and Banerjee (2012).
The reason for the above change is as follows. In the present paper, we need to emphasise that the test statistics are functions of Y. Besides, it is important to note that , as stated in Hu and Banerjee (2012).
In the next section, we show that some results, which are more accurate and stronger than (b), hold true.

THE MAIN THEOREM
The main objective of this paper is to prove that the strict inequalities hold among the powers of the three test statistics . Thus the assumption of the theorem is quite natural.
The results of the theorem may seem to be intuitively obvious. But some careful geometrical discussions are needed to prove that those strict inequalities hold for all nequalities hold for all 0 t .
In order to prove the theorem, we state three lemmas. The proofs of these lemmas need some geometrical discussions and they are given in the Appendix.

Lemma 1.
for any , all , and all and , and all , . Thus we complete the proof of the first inequality.
Next we prove the second inequality of the theorem. Let when ) (Y is and any positive number t be fixed again. Let ℬ be the non-empty set in R p×q defined in Lemma 3. Then we can apply exactly the same technique as that in the proof of the first inequality to complete the proof. □

CONCLUSION
We considered the problem of testing homogeneity of regression coefficient vectors under synchronised order restrictions when the covariance matrices are common but unknown. For this problem, Hu and Banerjee (2012) proposed some test statistics, studied their distributions, and obtained some inequalities among their powers. In the present paper, we mathematically proved that the strict inequalities hold among the powers of the test statistics through a careful geometrical discussion. Thus we have attained an exact comparison among the powers of the test statistics indicating more accurate and stronger results than those obtained in Hu and Banerjee (2012).

March 2016
Journal of the National Science Foundation of Sri Lanka 44(1)

APPENDIX: PROOFS OF LEMMAS
Here we state the following lemma in order to prove Lemmas 1, 2 and 3.

Lemma 4.
For any H, a non-empty subset of , a non-empty subset of } ,..., 2 , 1 { p , t , the following statements are true.
(a) . (c): by using the relation between the norm and inner product defined in Preliminaries. By equation (5) in Hu and Banerjee (2012), also by equation (5) in Hu and Banerjee (2012). Hence so the second equality of (c) follows. Further, by (a) and (b), the last equality of (c) follows.
because of (c) and the fact that because of (c) and the fact that Thus (d) follows. □ Proof of Lemma 1.