论文信息 - The asymptotic minimax constant for sup-norm loss in nonparametric density estimation

The asymptotic minimax constant for sup-norm loss in nonparametric density estimation

We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has already been found for nonparametric regression and for signal estimation in Gaussian white noise. H61der classes for arbitrary smoothness index > 0 on the unit interval are considered. The constant involves the value of an optimal recovery problem as in the white noise case, but in addition it depends on the maximum of densities in the function class.

M. Nussbaum | A. Korostelev

[1] Charles A. Micchelli,et al. A Survey of Optimal Recovery , 1977 .

[2] T. J. Rivlin,et al. Optimal Estimation in Approximation Theory , 1977 .

[3] O. V. Besov,et al. Integral representations of functions and imbedding theorems , 1978 .

[4] R. Khasminskii. A Lower Bound on the Risks of Non-Parametric Estimates of Densities in the Uniform Metric , 1979 .

[5] C. J. Stone,et al. Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[6] I. A. Ibragimov,et al. Bounds for the Risks of Non-Parametric Regression Estimates , 1982 .

[7] M. Freidlin,et al. Random Perturbations of Dynamical Systems , 1984 .

[8] Luc Devroye,et al. Nonparametric Density Estimation , 1985 .

[9] James V. Bondar,et al. Mathematical theory of statistics , 1985 .

[10] J. Wellner,et al. Empirical Processes with Applications to Statistics , 2009 .

[11] L. Devroye,et al. Nonparametric density estimation : the L[1] view , 1987 .

[12] A. D. Wentzell,et al. Limit Theorems on Large Deviations for Markov Stochastic Processes , 1990 .

[13] D. Donoho. Asymptotic minimax risk for sup-norm loss: Solution via optimal recovery , 1994 .

[14] M. Nussbaum. Asymptotic Equivalence of Density Estimation and Gaussian White Noise , 1996 .

[15] Cun-Hui Zhang,et al. Asymptotic nonequivalence of nonparametric experiments when the smoothness index is , 1998 .