Some remarks on the testing of smooth changes in the linear drift of a stochastic process (Q2740459)

One of the most extensive areas in change-point analysis is that of testing for a change in the mean of a sequence of observations. From a practical point of view, cases of dependent observations have become more and more important. A key tool in developing suitable change-point tests for the latter situation consists in embedding the observed sequence in a Gaussian framework and then carry out the statistical analysis (via invariance) in the corresponding asymptotic model. \textit{L. Horváth} and the present author [J. Stat. Plann. Inference 91, No. 2, 365-376 (2000)] pursued this idea in a rather general model of a stochastic process with an ``abrupt'' change in the mean (or variance) of its increments. \textit{D. Jarušková} [J. Stat. Plann. Inference 70, No. 2, 263-276 (1998; Zbl 0938.62071)] and \textit{M. Hušková} [B. Szyszkowicz (ed.), Asymptotic methods in Probability and Statistics, 577-583 (1998; Zbl 0945.62023)] proposed some testing procedures for a location model with a ``gradual'' (linear) change in the mean of independent observations. These testing procedures have been extended by \textit{M. Hušková} and the author [J. Stat. Plann. Inference 89, No.1-2, 52-77 (2000)] to cover a whole class of change alternatives.NEWLINENEWLINENEWLINEThe main aim of this paper is to show that the corresponding analysis of gradual changes can be carried out in a rather general model. Namely, some weighted approximations are developed resulting in limiting null distributions for a class of statistics testing a ``smooth'' change in the linear drift of a stochastic process.NEWLINENEWLINENEWLINEThe results extend those of the above-mentioned on testing of gradual changes in a location model with i.i.d. errors to cover more general classes of stochastic processes satisfying a certain weak invariance principle. Examples included are partial sums of independent observations with gradual changes in the mean, renewal processes with a smooth change in the drift and also linear processes from time series analysis. Consistency of the suggested test procedures is briefly discussed.