On Two $K$-Sample Rank Tests for Censored Data

Abstract

Let $X_{ij} (j = 1, 2,\cdots, n_i; i = 1, 2,\cdots, k)$ be $k$ independent samples of sizes $n_1, n_2,\cdots, n_k$ respectively from $k$ populations with continuous cumulative distribution functions $F_1, F_2,\cdots, F_k$ respectively. We assume that the $F_i$'s belong to a family $\mathscr{F}$ of distribution functions indexed by a parameter $\theta$. Let all the $N = \sum^k_{i=1} n_i$ observations be put together and ordered to form a single sequence and suppose that only the first $r$ ordered observations are available. That is, let us have a combined (right) censored sample of total size $r$. Such a censored sample occurs naturally in many physical situations as for example, in problems of life testing where we are interested in comparing the mean life of several physical systems, or in clinical trials or bio-assay problems where we want to compare the efficacy of several drugs but we cannot afford to wait indefinitely to get information on all the sampling units put on test. For details see Basu [2]. For facility of discussion, we shall use the terminology of life testing. Any test based on the first $r$ ordered observations (out of a combined sample of size $N$) will be termed an $r$ out of $N$ test. In this paper we propose two $k$-sample $r$ out of $N$ rank tests which generalize the rank tests proposed by Kruskal [7], Jonckhere [6] and Terpstra [10], [11]. In the first part of the paper we propose the statistic $B_r^{(N)}$ (large values being critical) to test the null hypothesis \begin{equation*}\tag{1.1}H_0:F_1(x) = F_2(x) = \cdots = F_k(x)\end{equation*} (or equivalently, $H_0:\theta_1 = \theta_2 = \cdots = \theta_k = 0$ say, under location alternatives) against the alternative hypothesis \begin{equation*}\tag{1.2}H_1:F_i(x) = F(x, \theta_i) \quad (i = 1, 2, \cdots, k)\end{equation*} In Section 2 we define the statistic $B_r^{(N)}$ and show its relationship with other statistics. The mean and variance of $B_r^{(N)}$ under the null hypothesis is derived in Section 3. In Section 4 we find the asymptotic distribution of $B_r^{(N)}$ both under the null and the non-null case. The computation of $B_r^{(N)}$ has been illustrated by an example in Section 5. In the second part of the paper we consider the statistic $V(N, r)$ (to be defined later) for testing the hypothesis (1.1) against the ordered alternative \begin{equation*}\tag{1.3}H_2:F_1(x) < F_2(x) < \cdots < F_k(x)\end{equation*} for all $x$, where the $F_i(x)$ are labeled in such a way that (1.3) is the ordered alternative being considered. The second formulation is useful in life testing where the $k$ cdf's might be associated with $k$ different processes and the experimenter wishes to test whether the $k$ processes give rise to units with the same life-time distributions against the alternative that the processes can be ordered in a particular manner, in the sense that the $k$ life-time distributions can be ordered uniformly with respect to $x$, (that is, according to their reliabilities). In Section 6 we define the statistic $V(N, r)$ and in Section 7 we have derived the mean and variance of $V(N, r)$ under $H_0$. Section 8 is devoted to investigate the extreme values of $V(N, r)$ and the asymptotic normality of $V(N, r)$ is proved in Section 9. Both the statistics $B_r^{(N)}$ and $V)N, r)$ may be considered as $k$-sample extensions of the $V_r$ statistic proposed by Sobel [9]. The definitions of $B_r^{(N)}$ and $V(N, r)$ are slightly different from what is given earlier in Basu [2].