Minimax Results for IFRA Scale Alternatives

Abstract

Recent results by Birnbaum, Esary and Marshall (1966), Barlow and Proschan (1967) and others suggest that the exponential models used be Epstein and Sobel (1953) and others for life testing problems should be extended to models in which the lifetimes have increasing failure rate average (IFRA) distributions. In this paper, IFRA scale models are dealt with. Consider two independent random samples $X_1, \cdots, X_m$ and $Y_1, \cdots, Y_n$ from populations with distributions $F(x)$ and $G(y) = F(\Delta y)$ respectively, where $F$ is a continuous, unknown, IFRA distribution. The null hypothesis $H_0:\Delta \leqq 1$ is to be tested against the alternative $H_1:\Delta > 1$. Since $F$ is unknown, it is not possible to maximize the power $E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack, \Delta > 1$. However, it is shown (Theorems 2.1, 2.2, 3.1, 3.2) that the tests that maximize the power for the exponential alternative $F(\Delta y) = 1 - \exp (- \Delta y)$ actually maximize the minimum power $\inf_{F(\cdot)} E\lbrack \phi\mid F(\cdot), F(\Delta\cdot) \rbrack$. Thus these tests are minimax. They have been computed by Lehmann (1953), Savage (1956), and Rao, Savage and Sobel (1960). The results indicate that in the case of uncensored samples, one should use one of the statistics \begin{align*}L = {\prod}^n_{i=1} (N + i - s_{n+1-i}), \text{or} \\ S = \sum^m_{i=1} J_0(r_i), \text{with} J_0(k) = \sum^N_{j=N+1-k} 1/j,\end{align*} where $N = m + n$ and $r_1, \cdots, r_m (s_1, \cdots, s_n)$ are the ordered ranks of the $X$'s ($Y$'s) in the combined sample of $X$'s and $Y$'s. $L$. is minimax for $\Delta$ in an interval about two, and $S$ is minimax for $\Delta$ in an interval $(1, \delta)$ to the right of one. The minimax statistics in the case of censored samples are more complicated (see (3.1) and (3.2)) and one might use one of the approximations suggested by Gastwirth (1965) or Basu (1967) (see (3.3)). Only finite sample size properties are dealt with. Asymptotic results are given in [6].