Density Estimation for the Case of Supersmooth Measurement Error

Abstract

The problem is to estimate the probability density of a random variable contaminated by an independent measurement error. I explore one of the worst-case scenario when the characteristic function of this measurement error decreases exponentially and thus optimal estimators converge only with logarithmic rate. The particular example of such measurement error is any random variable contaminated by normal, Cauchy, or another stable random variable. For this setting and circular data, I suggest an asymptotically efficient data-driven estimator that is adaptive to both smoothness of estimated density and distribution of measurement error. Moreover, this estimator is universal in sense that its derivatives and integral are sharp estimators of the corresponding derivatives and the cumulative distribution function, and these estimators are sharp both globally and pointwise. For the case of small sample sizes, I suggest a modified estimator that mimics an optimal linear pseudoestimator. I explore this estimator theoretically, via intensive Monte Carlo simulations and practical examples.