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The asymptotic behavior of some nonparametric change-point estimators. (English) Zbl 0776.62032
The author studies the following change-point problem in a random seminorm framework: Let $$X_ 1,X_ 2,\dots,X_ n$$ be a sequence of independent random variables, where $$X_ 1,X_ 2,\dots,X_{n\theta}$$ have distribution $$P$$, and $$X_{n\theta+1},X_{n\theta+2},\dots,X_ n$$ have distribution $$Q$$. The author proceeds to show that the nonparametric estimators proposed by E. Carlstein [ibid. 16, No. 1, 188-197 (1988; Zbl 0637.62041)] and B. S. Darkhovskij [Theory Probab. Appl. 21, 178-183 (1976); translation from Teor. Verojatn. Primen. 21, 180-184 (1976; Zbl 0397.62024)] are consistent with rate $$O_ p(n^{-1})$$. The improvement of the convergence rate is based on Hotelling’s exponential bound for a VC class. He derives the limiting distributions of certain estimators corresponding to four particular seminorms. The limiting distributions are determined by way of some functionals of two-sided Brownian motion with drift. He also compares the efficiencies of nonparametric estimators relative to some semiparametric estimators. Finally, the author proposes a bootstrap procedure for constructing a confidence set for the change point and then applies it to the Nile data.
Reviewer: G.Roussas (Davis)

##### MSC:
 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference 62G09 Nonparametric statistical resampling methods 62E20 Asymptotic distribution theory in statistics 62G15 Nonparametric tolerance and confidence regions
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