Estimation for Monotone Parameter Sequences: The Discrete Case

Abstract

Let $x_1, x_2, \cdots$ be a sequence of independent, observable, discrete, real random variables with respective density functions $f(x, \theta_1), f(x, \theta_2), \cdots$, where $\theta_i\in\Omega$ (a real interval), $i = 1,2, \cdots$. At any stage $s$, having observed $x_1, \cdots, x_s$, we wish to estimate $\theta_s$, relative to the loss function $L(d, \theta) \geqq 0$, under the restraint that $\theta_1 \leqq \cdots \leqq \theta_s$. Most of the literature concerned with estimating ordered parameters deals with maximum likelihood estimators (for example, Robertson and Waltman [3] and references contained therein). Recently, Blumenthal and Cohen [1] considered questions of minimaxity and admissibility of certain estimators of ordered translation parameters of continuous distributions. In Section 3 we obtain results concerning admissible and minimax estimators of $\theta_s$ for a large class of discrete distributions. Theorem 3.2 states that the minimax value $M_k$ of the problem of estimating $\theta_k$ at stage $k$ is the same for all $k = 1,2,\cdots$. Furthermore, the Corollary to Theorem 3.1 states that if $t(x_1, \cdots, x_s)$ is admissible for estimating $\theta_s$ at stage $s$, then $t(x_{k+1}, \cdots, x_{k+s})$ is admissible for estimating $\theta_{k+s}$ at stage $k + s$. This is not true, for example, when $x_k$ is normally distributed with mean $\theta_k$ and unit variance [4]. Hence in situations satisfying the conditions of Theorems 3.1 and 3.2, if there exists an admissible estimator $t(x_1)$ of $\theta_1$ having constant risk, then $t(x_s)$ will be the unique admissible minimax estimator of $\theta_s$, at stage $s$. It is this undesirable property (i.e. being based only on the last observation) which prompts us to look for other estimators which may be "better" in some sense. In Section 4 we use the early observations $(x_1, \cdots, x_{s-1}$, at stage $s$) to construct a sequence of estimators which is asymptotically subminimax $(s \rightarrow \infty)$ as well as having desirable properties for finite $s$. The results in Section 4 are obtained for general cumulative distribution functions as the methods are not peculiar to the discrete situation. However, it should be noted that the main results of Section 4 (Theorem 4.2 and Theorem 4.3) do not yield a worthwhile sequence of estimators unless we are in a situation similar to that of Section 3. That is, unless the existing admissible minimax estimators are undesirable as they are in the discrete case of Section 3.