Maximum likelihood ratio test for the stability of sequence of Gaussian random processes

Based on the Karhunen–Loeve expansion, the maximum likelihood ratio test for the stability of sequence of Gaussian random processes is investigated. The likelihood function is based on the first pp scores of eigenfunctions in the Karhunen–Loeve expansion for Gaussian random processes. Though the scores are unobservable, we show that the effect of the difference between scores and their estimators is negligible as the sample size tends to infinity. The asymptotic distribution is proved to be the Gumbel extreme value distribution. Under the alternative the test is shown to be consistent. For different choices of pp, simulation results show that the test behaves quite well in finite samples. The test procedure is also applied to the annual temperature data of central England. The results show that the temperatures have risen in the last twenty years, however there is no evidence to show that the autocovariance functions of the temperatures have changed among the range of the observations.