On approximation of the level probabilities and associated distributions in order restricted inference
Tim Robertson, Farroll T. Wright · Biometrika · 1983
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.