Likelihood Ratio Test for a Change Point in a Sequence of Independent Exponentially Distributed Random Variables

摘要

Let x1, ..., xn + 1 be independent exponentially distributed random variables, and let xi have intensity lambda 1 for i > or = tau, and intensity lambda 2 for i > tau, where tau is an unknown instant and lambda 1 and lambda 2 are also unknown. It is proved that the asymptotic null-distribution of the likelihood ratio statistic for testing lambda 1 = lambda 2 (or, equivalently, tau = 0 or n + 1) is an extreme value distribution, by application of theorems concerning the normed uniform quantile process. The rate of convergence is studied with Monte Carlo methods. Since it appears very low, simulated 5% critical values are given. It is shown that the test is optimal in the sense of Bahadur. Simulation results indicate a good power for values of n that are relevant for most applications. The likelihood ratio test is compared with a test which has the same asymptotic null-distribution. It is proved that this test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test.