## The Annals of Mathematical Statistics

### Estimation for Monotone Parameter Sequences: The Discrete Case

Harold Sackrowitz

#### 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  and references contained therein). Recently, Blumenthal and Cohen  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 . 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.

#### Article information

Source
Ann. Math. Statist., Volume 41, Number 2 (1970), 609-620.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177697101

Digital Object Identifier
doi:10.1214/aoms/1177697101

Mathematical Reviews number (MathSciNet)
MR254961

Zentralblatt MATH identifier
0195.19901

JSTOR