Deconvolution of a Cumulative Distribution Function with Some Non-standard Noise Densities

Let X be a continuous random variable having an unknown cumulative distribution function F. We study the problem of estimating F based on i.i.d. observations of a continuous random variable Y from the model Y = X + Z. Here, Z is a random noise distributed with known density g and is independent of X. We focus on some cases of g in which its Fourier transform can vanish on a countable subset of ℝ. We propose an estimator $\hat F$ for F and then investigate upper bounds on convergence rate of $\hat F$ under the root mean squared error. Some numerical experiments are also provided.