Analysis of a monotone trend in a multiparameter case

The problem of analyzing a monotone trend is considered. An estimate of the maximum likelihood of distribution parameters is built when the monotonicity condition is formulated for the values of some function from them. The solution of the corresponding problem is obtained in the form of an algorithm which generalizes the PAV (Pool-Adjacent-Violators) procedure. As an example, the problem of estimating the monotone trend of the ratio of mathematical expectation to the standard for a sequence of normally distributed quantities is considered. The resulting estimate is based on a count of the number of positive / negative values observed. It is shown that trend testing in this case is equivalent to the analysis of monotone changes in the probability of success in a heterogeneous Bernoulli scheme. Thus, the connection between the parametric and nonparametric approaches in the analysis of nonstationary random sequences is revealed. An example of a real situation when it is possible to apply the approach under consideration is the analysis of random sequences in a transformed form: a set of observations is divided into groups, for each of which some statistics are calculated, the result of such fragmentation is considered as a sequence of values with a certain distribution.