Estimation of Starshaped Sequences of Poisson and Normal Means

Abstract

A vector $\mu = (\mu_1, \cdots, \mu_n)$ is said to be upper [lower] starshaped if $\mu_{m + 1} \geqslant 0 \lbrack \leqslant \mu_{m + 1} \leqslant \bar{\mu}_m\rbrack m = 1, \cdots, n - 1$, where $\bar{\mu}_m$ is a weighted average of $\mu_1, \cdots, \mu_m$. Obtained is the maximum likelihood estimate of $\mu$ when the $\mu_i$'s are the means of $n$ Poisson or normal populations and $\mu$ is known to be starshaped. The method is applied to obtain estimators of IHRA (increasing hazard rate average) distribution functions.