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

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.

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 |