The law of the iterated logarithm for the integrated squared deviation of a kernel density estimator. (English) Zbl 1067.62048
Summary: Let \(f_{n,K}\) denote a kernel estimator of a density \(f\) in \(\mathbb{R}\) such that \(\int_\mathbb{R} f^p(x)dx<\infty\) for some \(p>2\). It is shown, under quite general conditions on the kernel \(K\) and on the window sizes, that the centred integrated squared deviation of \(f_{n,K}\) from its mean,
\[
\|f_{n,K}-Ef_{n,K} \|^2_2-E\|f_{n,K}-Ef_{n,K}\|^2_2,
\]
satisfies a law of the iterated logarithm (LIL). This is then used to obtain a LIL for the deviation from the true density, \(\|f_{n,K}-f\|^2_2-E\|f_{n,K}-f\|^2_2\).
The main tools are the J. Komlós, P. Major and G. Tusnády approximation, [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975; Zbl 0308.60029)], a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work of M. Pinsky [Proc. Am. Math. Soc. 22, 288–290 (1969; Zbl 0185.46601)], and an exponential inequality due to E. Giné, R. Latala and J. Zinn [E. Giné et al. (eds), High-Dimensional Probability II, Prog. Probab. 47, 13–38 (2000; Zbl 0969.60024)], for degenerate \(U\)-statistics applied in combination with decoupling and maximal inequalities.
The main tools are the J. Komlós, P. Major and G. Tusnády approximation, [Z. Wahrscheinlichkeitstheor. Verw. Geb. 32, 111–131 (1975; Zbl 0308.60029)], a moderate-deviation result for triangular arrays of weighted chi-square variables adapted from work of M. Pinsky [Proc. Am. Math. Soc. 22, 288–290 (1969; Zbl 0185.46601)], and an exponential inequality due to E. Giné, R. Latala and J. Zinn [E. Giné et al. (eds), High-Dimensional Probability II, Prog. Probab. 47, 13–38 (2000; Zbl 0969.60024)], for degenerate \(U\)-statistics applied in combination with decoupling and maximal inequalities.
MSC:
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G07 | Density estimation |
| 62F15 | Bayesian inference |
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