Electronic Journal of Statistics

Deconvolution for an atomic distribution

Bert van Es, Shota Gugushvili, and Peter Spreij

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Let X1,,Xn be i.i.d. observations, where Xi=Yi+σZi and Yi and Zi are independent. Assume that unobservable Y’s are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Zi have the standard normal distribution and let σ>0. Based on a sample X1,,Xn, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

Article information

Electron. J. Statist., Volume 2 (2008), 265-297.

First available in Project Euclid: 30 April 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62G20: Asymptotic properties

Asymptotic normality atomic distribution deconvolution kernel density estimator


van Es, Bert; Gugushvili, Shota; Spreij, Peter. Deconvolution for an atomic distribution. Electron. J. Statist. 2 (2008), 265--297. doi:10.1214/07-EJS121. https://projecteuclid.org/euclid.ejs/1209565146

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