Distributions of likelihood ratio statistics for monotone trend detection

The problem of analyzing a monotonic trend is considered. The model is based on the change in the mean of the normal distribution, for which the distributions of the likelihood ratio statistics are mixtures of chi-square or beta distributions with weights determined through the Stirling numbers of the first kind. The questions of approximation of these distributions are investigated. An effective method for calculating significance criteria by discarding insignificant elements of a mixture is proposed which makes it possible to reduce the complexity of calculations by several orders of magnitude. The issues of expanding the criteria for analyzing the trend of the mean normal distribution to procedures for identifying stochastic ordering are discussed. For these purposes, the authors apply the monotonic trend criterion for the averaged ranks, the statistics of which, in the absence of changes, is distributed as a mixture of chi-square distributions. The use of a graded assessment structure was of fundamental importance: it was possible not only to obtain original results, but also to ensure the viability of the proposed solutions.