Maximum Likelihood Estimates of Monotone Parameters

The maximum likelihood estimators of distribution parameters subject to certain order relations are determined for simultaneous sampling from a number of populations, when (i) the order relations may be specified by regarding the distribution parameters, of which one is associated with each population, as values at specified points of a function of n variables (n a positive integer), monotone in each variable separately; (ii) the distributions of the populations from which sample values are taken belong to the exponential family defined below. This family includes, in particular, the binomial, the normal with fixed standard deviation and variable mean, the normal with fixed mean and variable standard deviation, and the Poisson distributions.