Estimating Ordered Probabilities

Abstract

Let $X_i, Y_i i = 1, 2, \cdots n$ be mutually independent binomial random variables with $P\{X_i = 1\} = p_1, P\{Y_i = 1\} = p_2$. Let $\bar X_n = (1/n) \sum^n_{i = 1} X_i$ and $\bar Y_n = (1/n) \sum^n_{i = 1} Y_i$. We wish to estimate the parameter $p = (p_1, p_2)$. If the parameter space is $\Omega = \{p \mid 0 \leqq p_1 \leqq 1, 0 \leqq p_2 \leqq 1\}$ then the usual non-sequential estimator $\delta = (\delta_1, \delta_2)$ is of the form $(f(\bar X_n)), f(\bar Y_n))$, that is, if no restriction is placed on the parameters, the estimator for the paired parameters is the pair of estimators for each parameter separately. In this paper, however, we are concerned with the parameter space $\Omega = \{p \mid 0 \leqq p_1 \leqq p_2 \leqq 1\}$. We show that estimators constructed as before are no longer admissible with respect to a class of reasonable loss functions. Square error loss is included in this class. In particular, we show that for such loss functions, estimators not ordered in the same way as the parameters are inadmissible. A class of estimators which retains the same ordering as the parameters, that is with $\delta_2 \geqq \delta_1$, is investigated and the asymptotic behavior of the minimax member is described. Finally, an asymptotic estimator based on a normal approximation is given. This estimator is minimax and admissible. The problem of estimating ordered probabilities arises in estimating a section of an unknown distribution function.