On Estimating Monotone Parameters

Abstract

Suppose for each $i = 1,2, \cdots, k$ the random variable $X_i$ has density function $f(x; \theta_i)$ where each of the parameters $\theta_1, \theta_2, \cdots, \theta_k$ is known to belong to some connected set $\Theta$ of real numbers. Independent random samples are taken from the distributions of $X_1, X_2, \cdots, X_k$ and we wish to find estimates $\hat{\theta}_1, \hat{\theta}_2, \cdots, \hat{\theta}_k$ of $\theta_1, \theta_2, \cdots, \theta_k$ which satisfy: \begin{equation*}\tag{1.1}\hat{\theta}_1 \geqq \hat{\theta}_2 \geqq \cdots \geqq \hat{\theta}_k.\end{equation*} Brunk [3] considered such a problem when $f(x; \theta)$ belongs to a certain exponential family of distributions which includes the binomial, the normal with fixed mean and variable standard deviation, the normal with fixed standard deviation and variable mean, and the Poisson distributions. A discussion of the history of this type of problem is given by Brunk [4]. In this paper we assume that the density function $f(x; \theta)$ has certain properties and develop a procedure for finding the restricted estimates. The above mentioned densities have those properties as do certain others. One in particular, not covered by Brunk's formulation, the bilateral exponential distribution (i.e. $f(x; \theta) = \frac{1}{2}e^{-|x-\theta|}$), is considered in Section Four. In Section Two we list those assumptions, describe our procedure for finding restricted estimates and prove that they are maximum likelihood estimates. In Section Three we give a representation theorem for our estimates and a theorem which implies that they are consistent in the special cases which we consider. In Section Five we describe an alternate method for obtaining restricted estimates and in Section Six we discuss another special case not considered by Brunk [3], which does not satisfy our regularity assumptions but for which the procedure described in Section Two clearly works.