The likelihood ratio test for the change point problem for exponentially distributed random variables

Let x1,..., xn+1 be independent exponentially distributed random variables with intensity [lambda]1 for i [less-than-or-equals, slant] [tau] and [lambda]2 for i> [tau], where [tau] as well as [lambda]1 and [lambda]2 are unknown. By application of theorems concerning the normed uniform quantile process 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. Change point problems occur in a variety of experimental sciences and therefore have considerabla attention of applied statisticians. The problems are non-standard since the usual regularity conditions are not satisfied. Explicit asymptotic distributions of likelihood ratio tests have until now only been derived for a few cases. The method of proof used in this paper is based on the 'strong invariance principle'. Furthermore it is shown that the test is optimal in the sense of Bahadur, although the Pitman efficiency is zero. However, simulation results indicate a good power for values of n that are relevant for most applications. The likelihood ratio test is compared with another test which has the same asymptotic null-distribution. This test has Bahadur efficiency zero. The simulation results confirm that the likelihood ratio test is superior to the latter test.