Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator.Comment: Published at http://dx.doi.org/10.3150/07-BEJ6091 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Let $X_1,...,X_n$ be i.i.d. observations, where $X_i=Y_i+\sigma Z_i$ and $Y_i$ and $Z_i$ 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 $Z_i$ have the standard normal distribution and let $\sigma>0.$ Based on a sample $X_1,..., X_n,$ 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.Comment: Published in at http://dx.doi.org/10.1214/07-EJS121 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Abstract. Let X 1 , . . . , Xn be i.i.d. copies of a random variable X = Y + Z, where X i = Y i + Z i , and Y i and Z i are independent and have the same distribution as Y and Z, respectively. Assume that the random variables Y i 's are unobservable and that Y = AV, where A and V are independent, A has a Bernoulli distribution with probability of success equal to 1 − p and V has a distribution function F with density f. Let the random variable Z have a known distribution with density k. Based on a sample X 1 , . . . , Xn, we consider the problem of nonparametric estimation of the density f and the probability p. Our estimators of f and p are constructed via Fourier inversion and kernel smoothing. We derive their convergence rates over suitable functional classes. By establishing in a number of cases the lower bounds for estimation of f and p we show that our estimators are rate-optimal in these cases.