On a Generalized Savage Statistic with Applications to Life Testing

Abstract

Let there be two samples $X_1, X_2, \cdots, X_m$ and $Y_1, Y_2, \cdots, Y_n(N = m + n)$ from two populations with continuous cdf's $F(x)$ and $G(y)$. Let the first $i$ ordered observations (out of $N$ combined observations) contain $m_ix$'s and $n_iy$'s $(m_i + n_i = i)$ where $m_i$ and $n_i$ are random numbers. To test \begin{equation*}\tag{0.1}H_0:F = G\end{equation*} against alternative that they are different we propose the statistic \begin{equation*}\tag{0.2}S^{(N)}_r = \sum^r_{i = 1} a_iz_i + (m - m_r)(N - r)^{-1}(\sum^N_{r+1}a_i) - \frac{1}{2}(m + n)\end{equation*} based on the first $r$ ordered observations only where $a^N_i = a_i = \sum^N_{j=N-i+1} 1/j,$ and \begin{align}z_i &= 1, \text{if the} i\text{th ordered observation is an} x_i, \\ &= 0, otherwise.\end{align} The statistic is the asymptotically most powerful rank test for censored data under the Lehmann alternative and is equivalent to the Savage statistic [14] when $r = N$. It is also known to maximize the minimum power over IFRA (or IFR) distributions asymptotically. Exact and large sample properties of $S^{(N)}_r$ are studied and a $k$-sample extension of it is also considered. Various tables are also provided to facilitate the use of the $S^{(N)}_r$ statistic.