On the asymptotic behaviour of the integrated square error of kernel density estimators with data-dependent bandwidth. (English) Zbl 0982.62032

Tenreiro, Carlos

Stat. Probab. Lett. 53, No. 3, 283-292 (2001).

Summary: We consider the integrated square error \(J_n= \int\{ \widehat{f}_n(x)- f(x)\}^2\) dx, where \(f\) is the common density function of the independent and identically distributed random vectors \(X_1,\dots, X_n\) and \(\widehat{f}_n\) is the kernel estimator with a data-dependent bandwidth. Using the approach introduced by P. Hall [J. Multivariate Anal. 14, 1–16 (1984; Zbl 0528.62028)], and under some regularity conditions, we derive the \(L_2\) consistency in probability of \(\widehat{f}_n\) and we establish an asymptotic expansion in probability and a central limit theorem for \(J_n\).

MSC:

62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference
60F05	Central limit and other weak theorems

Keywords:

kernel estimators; integrated square error; asymptotic distribution; U-statistics

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