Umeå universitets logga

In this thesis, we study statistical inference for entropy, divergence, and related functionals of one or two probability distributions. Asymptotic properties of particular nonparametric estimators of such functionals are investigated. We consider estimation from both independent and dependent observations. The thesis consists of an introductory survey of the subject and some related theory and four papers (A-D).

In Paper A, we consider a general class of entropy-type functionals which includes, for example, integer order Rényi entropy and certain Bregman divergences. We propose U-statistic estimators of these functionals based on the coincident or epsilon-close vector observations in the corresponding independent and identically distributed samples. We prove some asymptotic properties of the estimators such as consistency and asymptotic normality. Applications of the obtained results related to entropy maximizing distributions, stochastic databases, and image matching are discussed.

In Paper B, we provide some important generalizations of the results for continuous distributions in Paper A. The consistency of the estimators is obtained under weaker density assumptions. Moreover, we introduce a class of functionals of quadratic order, including both entropy and divergence, and prove normal limit results for the corresponding estimators which are valid even for densities of low smoothness. The asymptotic properties of a divergence-based two-sample test are also derived.

In Paper C, we consider estimation of the quadratic Rényi entropy and some related functionals for the marginal distribution of a stationary m-dependent sequence. We investigate asymptotic properties of the U-statistic estimators for these functionals introduced in Papers A and B when they are based on a sample from such a sequence. We prove consistency, asymptotic normality, and Poisson convergence under mild assumptions for the stationary m-dependent sequence. Applications of the results to time-series databases and entropy-based testing for dependent samples are discussed.

In Paper D, we further develop the approach for estimation of quadratic functionals with m-dependent observations introduced in Paper C. We consider quadratic functionals for one or two distributions. The consistency and rate of convergence of the corresponding U-statistic estimators are obtained under weak conditions on the stationary m-dependent sequences. Additionally, we propose estimators based on incomplete U-statistics and show their consistency properties under more general assumptions.