GLOBAL BEHAVIOR OF DECONVOLUTION KERNEL ESTIMATES

The desire to recover an unknown density when the data are contaminated with errors leads to nonparametric deconvolution problems. The difficulty of deconvolution depends on both the smoothness of the error distribution and the smoothness of the prior. Under certain smoothness constraints, we show that deconvolution kernel density estimates achieve the best global rates of convergence n-k-l/2(k+beta)+1 under an L(p) (1 less-than-or-equal-to p < infinity) norm, where l is the order of the derivative function of the unknown density to be estimated, k is the degree of smoothness constraints, and beta is the degree of the smoothness of the error distribution. The results indicate that in the presence 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.