Estimating and Testing Trend in a Stochastic Process of Poisson Type

Abstract

Let $\{T_i: i = 1, 2, \cdots\}$ be a stochastic process of Poisson type, with $\lambda(t)$, the rate of occurrence of the events, depending on time. We may interpret $T_i$ as the time of occurrence of the $i$th event. In Section 2, starting with the joint density function of $T_1, \cdots, T_n$, the maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda (t) \leqq M$ being a non-decreasing function of time ($M$ some positive number) is found. In Section 3, starting with the conditional joint density of $T_1, \cdots, T_n$ given there are $n$ events in $(0, t^\ast\rbrack$, the conditional maximum likelihood estimate of $\lambda(t)$ subject to $0 \leqq \lambda(t)$ being a non-decreasing function of time is found. In Section 4, the conditional likelihood ratio test of the hypothesis that $\lambda(t)$ is constant against the alternate hypothesis that $\lambda(t)$ is not constant but is nondecreasing is found, and a limiting distribution is found which may be used to approximate the probability of a type I error for large sample size. Theorem 2.1 (Brunk-van Eeden), I believe is important in its own right. It is contained in the works of Brunk and van Eeden, although it is not explicitly stated. This theorem can be used as a basis for tests of hypotheses for constant parameters against increasing parameters or for increasing parameters against all other alternatives.