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A consistent estimator for spectral density matrix of a discrete time periodically correlated process

Azimmohseni M., Soltani A. R., Khalafi M., Ghalesary N. M.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

225--242

In this article, we introduce a weighted periodogram in the class of smoothed periodograms as a consistent estimator for the spectra density matrix of a periodically correlated process. We derive its li miting distribution that appears to be a certain finite linear combination of Wishart distribution. We also provide numerical derivations for our smoothed period ogram and exhibit its asymptotic consistency using simulated data.

Teoria informacji a statystyka matematyczna

Inglot T.

Mathematica Applicanda

2014

Vol. 42, No. 1

115--174

W niniejszym artykule przedstawiony jest zarys teorii informacji z probabilistycznego i statystycznego punktu widzenia. Ten nurt teorii informacji rozwijał się intensywnie w ostatnich dziesięcioleciach. Wpłynął tez w znaczący sposób na rozwój metod statystycznych. Celem artykułu jest wprowadzenie czytelnika w przystępny sposób w podana powyżej tematykę, dostarczenie mu pewnych intuicji i przybliżenie specyfiki podejścia teorio-informacyjnego w statystyce matematycznej.

In the paper we present an outline of the information theory from the probabilistic and statistical point of view. Such a direction of the information theory has been intensively developed in recent decades and significantly influenced a progress in the statistical methodology. The aim of the article is to introduce the reader into these problems, provide some intuitions and acquaint with a specific information-theoretic approach to the mathematical statistics. The first part of the paper is devoted to brief and easy of approach introduction to the main notions of the information theory like entropy, relative entropy (Kullback- Leibler distance), information projection and Fisher information as well as presentation of their most important properties including de Bruijn’s identity, Fisher information inequalities and entropy power inequalities. In the short second part we give applications of the notions and results from the first part to limit theorems of the probability theory such as the asymptotic equipartition property, the convergence of empirical measures in the entropy distance, large deviation principle with emphasis to Sanov theorem, the convergence of distributions of homogeneous Markov chains in the entropy distance and the central limit theorem. The main, last part of the article shows some most significant and important applications of the information theory to the mathematical statistics. We discuss connections of the maximum likelihood estimators with the information projections and the notion of sufficient statistic from the information-theoretic point of view. The problems of source coding, channel capacity and an amount of information provided by statistical experiments are presented in a statistical framework. Some attention is paid to the expansion of Clarke and Barron and its corollaries e.g. in density estimation. Next, applications of the information theory to hypothesis testing is discussed. We give the classical Stein’s Lemma and its generalization to testing composite hypothesis obtained by Bahadur and show their connections with the asymptotic efficiency of statistical tests. Finally, we briefly mention the problem of information criteria in a model seletion including the most popular two-stage minimal description length criterion of Rissanen. The enclosed literature is limited only to papers and books which are referred to in the paper.