On the distribution of the number of steps in an estimate of monotone trend

We consider a sequence of independent random variables whose probability distributions differ only by a one-dimensional parameter of the density and investigate the maximum likelihood estimate of a monotone trend of the parameter. Using the decomposability property of the estimate, we obtain new results on the distribution of the number of steps of this estimate in the case of discrete distributions. These results are given in a convenient form in terms of formal power series.