Asymptotically Optimal Statistics in Some Models with Increasing Failure Rate Averages

Abstract

Birnbaum, Esary and Marshall (1966) have shown that the class $\mathscr{F}$ of distributions with increasing failure rate averages $(IFRA)$ characterizes the concept of wear-out in the sense that $\mathscr{F}$ is the smallest class that contains the exponential distributions and is closed under the formation of coherent systems. In this note, statistical inference for models in which the distributions are unknown and $IFRA$ will be considered. Let $F$ and $G$ be defined by \begin{equation*}\tag{1.1}F(t) = H(t/\theta) \text{and} G(t) = H(t/\gamma)\end{equation*} where $H$ is an unknown $IFRA$ distribution with $H(0) = 0$. Then, for the two-sample problem where one tests the equality of the means of $F$ and $G$, it is shown that the Savage (1956) statistic maximizes the minimum power over $IFRA$ distributions asymptotically. This asymptotic minimax solution is extended to censored samples and it turns out that the Gastwirth (1965) modified version of the Savage statistic is asymptotically minimax for this case. Asymptotic uniqueness of these minimax solutions holds only in a class of rank tests. The results are extended to obtain an estimate of the ratio of the means that minimizes the maximum asymptotic variance over $IFRA$ distributions. Moreover, the results are shown to hold also for distributions with increasing failure rates $(IFR)$, extensions to the $k$-sample problem are given, and asymptotic efficiencies of the best test for exponential models are given.