p-values of a test on homogeneous means in a multivariate isotonic regression

This paper considers a multivariate isotonic regression model in which the columns of the mean matrix are constrained by a quasi-ordering. For the test on the homogeneity of the means an ad hoc test statistic is derived. When the order restriction is confined to the elements of only one row, another statistic is obtained. A sufficient condition is established under which the p-value of the test is expressed through the distribution of the second statistic at a specified parameter point. This result not only simplifies the expression of the p-values but also makes the computation for the p-values by the Monte Carlo method possible.