Test Procedures for Possible Changes in Parameters of Statistical Distributions Occurring at Unknown Time Points

Abstract

The present study is concerned with the properties of a test statistic proposed by H. Chernoff and S. Zacks [1] to detect shifts in a parameter of a distribution function, occurring at unknown time points between consecutively taken observations. The testing problem we study is confined to a fixed sample size situation, and can be described as follows: Given observations on independent random variables $X_1, \cdots, X_n$, (taken at consecutive time points) which are distributed according to $F(X; \theta_i); \theta_i \varepsilon \Omega$ for all $i = 1, \cdots, n$, one has to test the simple hypothesis: $H_0 : \theta_1 = \cdots = \theta_n = \theta_0$ ($\theta_0$ is known) against the composite alternative: $H_1 : \theta_1 = \cdots = \theta_m = \theta_0 \\ \theta_{m + 1} = \cdots = \theta_n = \theta_0 + \delta;\quad\delta > 0,$ where both the point of change, $m$, and the size of the change, $\delta$, are unknown $(m = 1, \cdots, n - 1), 0 < \delta < \infty$. A Bayesian approach led Chernoff and Zacks in [1] to propose the test statistic $T_n = \sum^{n - 1}_{i = 1} iX_{i + 1}$, for the case of normally distributed random variables. A generalization for random variables, whose distributions belong to the one parameter exponential family, i.e., their density can be represented as $f(x; \theta) = h(x) \exp \lbrack\psi_1(\theta)U(x) + \psi_2(\theta)\rbrack, \theta \varepsilon \Omega$ where $\psi_1(\theta)$ is monotone, yields the test statistic $T_n = \sum^{n - 1}_{i = 1} iU(x_{i + 1})$. In the present paper we study the operating characteristics of the test statistic $T_n$. General conditions are given for the convergence of the distribution of $T_n$, as the sample size grows, to a normal distribution. The rate of convergence is also studied. It was found that the closeness of the distribution function of $T_n$ to the corresponding normal distribution is not satisfactory for the purposes of determining test criteria and values of power functions, in cases of small samples from non-normal distributions. The normal approximation can be improved by considering the first four terms in Edgeworth's asymptotic expansion of the distribution function of $T_n$ (see H. Cramer [2] p. 227). Such an approximation involves the normal distribution, its derivatives and the semi-invariants of $T_n$. The goodness of the approximations based on such an expansion, and that of the simple normal approximation, for small sample situations, were studied for cases where the observed random variables are binomially or exponentially distributed. In order to compare the exact distribution functions of $T_n$ to the approximations, the exact forms of the distributions of $T_n$ in the binomial and exponential cases were derived. As seen in Section 4, these distribution functions are quite involved, especially under the alternative hypothesis. Tables of coefficients are given for assisting the determination of these distributions, under the null hypothesis assumption, in situations of samples whose size is $2 \leqq n \leqq 10$. For samples of size $n \geqq 10$ one can use the normal approximation to the test criterion and obtain good results. The power functions of the test statistic $T_n$, for the binomial and exponential cases, are given in Section 5. The comparison with the values of the power function obtained by the normal approximation is also given. As was shown by Chernoff and Zacks in [1], when $X$ is binomially distributed the power function values of $T_n$ are higher than those of a test statistic proposed by E. S. Page [5], for most of the $m$ values (points of shift) and $\delta$ values (size of shift). A comparative study in which the effectiveness of test procedures based on $T_n$ relative to those based on Page's and other procedures will be given elsewhere for the exponential case, and other distributions of practical interest.