Abstract
Let f n be the non-parametric kernel density estimator based on a kernel function K and a sequence of independent and identically distributed random variables taking values in ℝd. It is proved that if the kernel function is an integrable function with bounded variation, and the common density function f of the random variables is continuous and f(x) → 0 as |x| → ∞, then the moderate deviation principle and large deviation principle for \(\{ \sup _{x \in \mathbb{R}^d } |f_n (x) - E(f_n (x))|,n \geqslant 1\} \) hold.
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REFERENCES
Aubin, J. P. (1993). Optima and Equilibria, Springer, Berlin.
Dembo, A., and Zeitouni, O. (1998). Large Deviations Techmiques and Applications, Springer, New York.
Einmahl, U., and Mason, D. (1996). Some universal results on the behavior of increments of partial sums. Ann. Probab. 24, 1388–1407.
Einmahl, U., and Mason, D. (2000). An empirical process approach to the uniform consistency of kernel-type function estimators. J. Theor. Probab. 13, 1–37.
Ellis, R. S. (1984). Large deviations for a general class of random vectors. Ann. Probab. 12, 1–12.
Giné, E., and Guillou, A. (2001). On consistency of kernel density estimators for randomly censored data: Rates holding uniformly over adaptive intervals. Ann. Inst. H. Poincaré Probab. Statist. 37, 503–522.
Giné, E., and Guillou, A. (2001). Rates of strong uniform consistency for multivariate kernel density estimators. Ann. Inst. H. Poincaré Probab. Statist. 38, 907–921.
Giné, E., and Zinn, J. (1984). Some limit theorems for empirical processes. Ann. Probab. 12, 929–989.
Louani, D. (1998). Large deviations limit theorems for the kernel density estimator. Scand. J. Statist. 25, 243–253.
Nolan, D., and Pollard, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15, 780–799.
Silverman, B. W. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Statist. 6, 177–189.
Sion, M. (1958). On general minimax theorem. Pacific J. Math. 8, 171–176.
Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126, 505–563.
Wu, L. (1994). Large deviations, moderate deviations and LIL for empirical processes. Ann. Probab. 22, 17–27.
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Gao, F. Moderate Deviations and Large Deviations for Kernel Density Estimators. Journal of Theoretical Probability 16, 401–418 (2003). https://doi.org/10.1023/A:1023574711733
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DOI: https://doi.org/10.1023/A:1023574711733