Nonparametric Change-Point Estimation for Data from an Ergodic Sequence

Abstract

In the framework of the series scheme we assume that an observations sequence $\{ X_i^n , 1 \leqq i \leqq n\} $ is such that $X_i^n = U_i I(1 \leqq i \leqq [\theta n]) + V_i I([\theta n] + 1 \leqq i \leqq n)$, where $(U_i ,V_i )$ is a stationary ergodic sequence the marginal distributions of which are different, and $\theta $ is a change-point in the probabilistic characteristics such that $\theta \in (0;1)$. The main result of this paper is the proof of the fact that the sequence $(\theta n)_{n \geqq 1} $ of nonparametric estimations constructed here is consistent $(\theta n \to \theta )$.

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