Simple kernel estimators for certain nonparametric deconvolution problems

We consider deconvolution problems where the observations are equal in distribution to X = [lambda]1E1 + ... + [lambda]mEm + Y, or to X = [mu]1L1 + ... + [mu]mLm + Y. Here the random variables in the sums are independent, the Ei are exponentially distributed, the Li are Laplace distributed and Y has an unknown distribution F which we want to estimate. The constants [lambda]i and [mu]i are given. These problems include exponential, gamma and Laplace deconvolution. We derive inversion formulas, expressing F in terms of the distribution of the observations. Simple kernel estimators of F and its density f are then introduced by plugging in standard kernel estimators of the distribution of the observations. The pointwise asymptotic properties of the estimators are investigated.