On approximation of the level probabilities and associated distributions in order restricted inference

SUMMARY The use of much of the distribution theory developed for order restricted inference has been limited by the lack of algorithms for the 'level probabilities'. An approximation for these, which accounts for the pattern of large and small 'weights', is developed. This approximation and the equal weights approximation are examined. Both approximations appear to be reasonable for weight sets having a moderate amount of variability. The quality of the equal weights approximation, as a function of the amount of variability in the weights, deteriorates more quickly for certain patterns of large and small weights than for others. Thus, the approximation based upon the pattern of large and small weights is a significant improvement over the equal weights approximation. Finally, Siskind's (1976) approximation, which can be applied if the number of parameters is not too large, is discussed. The chi-bar-squared, %2, and E-bar-squared, P2, distributions are fundamental to the theory of order restricted hypothesis tests. These distributions have tail probabilities which are linear combinations of the tail probabilities of standard distributions and they depend upon the order,restriction through the coefficients in these linear combinations. The values of these coefficients are the probabilities that the order restricted maximum likelihood estimates of normal means assume specified numbers of distinct values, which are called levels. The estimates are based upon independent samples from each of the populations and depend upon the vector of relative precisions, called weights, of the sample means as estimates of the corresponding population means. The use of these tests has been limited because these level probabilities can be virtually impossible to compute if the weights are not all equal. In this paper we describe a technique for approximating the level probabilities for a linear order restriction and for unequal precisions. This approximation is based upon an idea of Chase (1974) and uses the pattern of large and small relative precisions. It seems to be particularly good when the relative precisions have two distinct values. However, it seems to provide a satisfactory approximation so long as the variation in the relative precisions is not too large. This is probably due to the lack of sensitivity of the level probabilities to changes in the weights. The robustness was