Communications of the
Korean Mathematical Society CKMS

In practice, the density function of a random variable $X$ is always unknown. Even its smoothness parameter is unknown to us. In this paper, we will consider a density smoothness parameter estimation problem via wavelet theory. The smoothness parameter is defined in the sense of equivalent Besov norms. It is well-known that it is almost impossible to estimate this kind of parameter in general case. But it becomes possible when we add some conditions (to our proof, we can not remove them) to the density function. Besides, the density function contains impurities. It is covered by some additive noises, which is the key point we want to show in this paper.

Keywords: smoothness parameter estimation, density, wavelets, additive noise, Besov spaces

MSC numbers: 62G05, 62G07, 42C40, 35Q30, 41A15