Some remarks on the testing of smooth changes in the linear drift of a stochastic process. (English. Ukrainian original) Zbl 0980.62076

Steinebach, Josef

Theory Probab. Math. Stat. 61, 173-185 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 164-175 (2000).

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. 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. D. Jarušková [J. Stat. Plann. Inference 70, No. 2, 263-276 (1998; Zbl 0938.62071)] and 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 M. Hušková and the author [J. Stat. Plann. Inference 89, No.1-2, 52-77 (2000)] to cover a whole class of change alternatives.
The 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.
The 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.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

62M07	Non-Markovian processes: hypothesis testing
62E20	Asymptotic distribution theory in statistics
60F17	Functional limit theorems; invariance principles
62F05	Asymptotic properties of parametric tests

Keywords:

change-point analysis; linear drift of stochastic processes; renewal processes; time series

Citations:

Zbl 0938.62071; Zbl 0945.62023