Nonparametric Tests for Shift at an Unknown Time Point

Abstract

This work is an investigation of a nonparametric approach to the problem of testing for a shift in the level of a process occurring at an unknown time point when a fixed number of observations are drawn consecutively in time. We observe successively the independent random variables $X_1, X_2, \cdots, X_N$ which are distributed according to the continuous cdf $F_i, i = 1, 2, \cdots, N$. An upward shift in the level shall be interpreted to mean that the random variables after the change are stochastically larger than those before. Two versions of the testing problem are studied. The first deals with the case when the initial process level is known and the second when it is unknown. In the first case, we make the simplifying assumption that the distributions $F_i$ are symmetric before the shift and introduce the known initial level by saying that the point of symmetry $\gamma_0$ is known. Without loss of generality, we set $\gamma_0 = 0$. Defining a class of cdf's $\mathscr{F}_0 = \{F:F$ continuous, $F$ symmetric about origin$\}$, the problem of detecting an upward shift becomes that of testing the null hypothesis $H_0:F_0 = F_1 = \cdots = F_N,\quad\text{some}\quad F_0 \varepsilon\mathscr{F}_0,$ against the alternative $H_1:F_0 = F_1 = \cdots = F_m > F_{m + 1} = \cdots = F_N,\quad\text{some}\quad F_0 \varepsilon\mathscr{F}_0$ where $m(0 \leqq m \leqq N - 1)$ is unknown and the notation $F_m > F_{m + 1}$ indicates that $X_{m + 1}$ is stochastically larger than $X_m$. For the second situation with unknown initial level, the problem becomes that of testing the null hypothesis $H_0^\ast:F_1 = \cdots = F_N$, against the alternatives $H_1^\ast: F_1 = \cdots = F_m > F_{m + 1} = \cdots = F_N$, where $m(1 \leqq m \leqq N - 1)$ is unknown. Here the distributions are not assumed to be symmetric. The testing problem in the case of known initial level has been considered by Page [11], Chernoff and Zacks [2] and Kander and Zacks [7]. Assuming that the observations are initially from a symmetric distribution with known mean $\gamma_0$, Page proposes a test based on the variables $\operatorname{sgn} (X_i - \gamma_0)$. Chernoff and Zacks assume that the $F_i$ are normal cdf's with constant known variance and they derive a test for shift in the mean through a Bayesian argument. Their approach is extended to the one parameter exponential family of distributions by Kander and Zacks. Except for the test based on signs, all the previous work lies within the framework of a parametric statistics. The second formulation of the testing problem, the case of unknown initial level, has not been treated in detail. The only test proposed thus far is the one derived by Chenoff and Zacks for normal distributions with constant known variance. In both problems, our approach generally is to find optimal invariant tests for certain local shift alternatives and then to examine their properties. Our optimality criterion is the maximization of local average power where the average is over the space of the nuisance parameter $m$ with respect to an arbitrary weighting $\{q_i, i = 1, 2, \cdots, N: q_i \geqq 0, \sum^N_{i = 1} q_i = 1\}$. From the Bayesian viewpoint, $q_i$ may be interpreted as the prior probability that $X_i$ is the first shifted variate. Invariant tests with maximum local average power are derived for the case of known initial level in Section 2 and for the case of unknown initial level in Section 3. In both cases, the tests are distribution-free and they are unbiased for general classes of shift alternatives. They all depend upon the weight function $\{q_i\}$. With uniform weights, certain tests in Section 3 reduce to the standard tests for trend while a degenerate weight function leads to the usual two sample tests. In Section 4, we obtain the asymptotic distributions of the test statistics under local translation alternatives and investigate their Pitman efficiencies. Some small sample powers for normal alternatives are given in Section 5.