Density Estimation in the Uniform Norm and White Noise Approximation
- Korostelev, Alexander P.
- Nussbaum, Michael
2010 Mathematics Subject Classification
- 62G07 62B15 62G20
- Density estimation, uniform norm loss, asymptotic minimax risk, exact constant, nonparametric experiments, Gaussian white noise, deficiency distance, asymptotic equivalence
We develop the exact constant of the risk asymptotics in the uniform norm for density estimation. This constant has first been found for nonparametric regression and for signal estimation in Gaussian white noise. We show that for densities with Hölder exponent > 1/2, the formal approximation of the i. i. d. experiment by Gaussian white noise in the sense of Le Cam's deficiency distance (asymptotic equivalence) can be utilized. For densities with Hölder exponent ≤ 1/2 where asymptotic equivalence fails, the result can still be established independently.