Global Behavior of Deconvolution Kernel Estimates

The desire to recover the unknown density when data are contaminated with errors leads to nonparametric deconvolution problems. The difficulty of deconvolution depends on both the smoothness of error distribution and the smoothness of the priori. Under a general class of smoothness constraints, we show that deconvolution kernel density k-l estimates achieve the best attainable global rates of convergence n 2(kH)+1 under L p (1 ~ p < 00) norm, where I is the order of the derivative function of the unknown density to be estimated, k is the degrees of smoothness constraints, and {3 is the degree of the smoothness of the error distribution. Our results indicate that in present of errors, the bandwidth should be chosen larger than the ordinary density estimate. These results also constitute an extension of the ordinary kernel density estimates. oAbbreviated title: global rates of deconvolution. AMS 1980 subject classification. Primary 62G20. Secondary 62G05.