Convergence rates of multivariate deconvolution and compound Poisson estimation

Xianglian Tang, Purdue University

Abstract

In this thesis, we tried to handle two problems: one is the multivariate deconvolution. We have seen many results in the literature about the univariate deconvolution, about different estimators and their consistency, their convergence rates; many results about the optimality of simple estimation, both parametrically and nonparametrically; even results about the optimality of univariate deconvolution problems. But no results concerning the optimality of multivariate deconvolution has appeared yet. We derived here the estimators, studied their convergence rates. And by deriving the lower bounds for the corresponding deconvolution problems, we established the optimality of kernel type estimators. With a little additional effort, we also studied the optimal properties of one univariate deconvolution cases in which the error term is lattice distributed. Another problem studied here is the convergence rates of maximum likelihood estimate of compound Poisson under Hellinger distance. As a special case of mixture problems, compound Poisson distribution has many application backgrounds, and the estimation of the mixing distribution has many practical meanings. A lot of researches dealing with estimation and their consistency has been done, but in the aspect of verification (in terms of convergence rate, for example), we have not seen many results. This paper also investigated the global behavior of the marginal distribution in terms of Hellinger distance when the mixing distribution is estimated by maximum likelihood method.

Degree

Ph.D.

Advisors

Loh, Purdue University.

Subject Area

Statistics

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