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Teoria informacji a statystyka matematyczna

Inglot T.

Mathematica Applicanda

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2014

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Vol. 42, No. 1

115--174

PL

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.

EN

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.

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Optimal method to determine density of the scarce vegetation in dry desert : Variable Area Transect (VAT) applied to saxaul (Haloxylon ammodenderon) community

Kiani B., Tabari M., Fallah A., Hosseini S., Iran Nejad Parizi M.

Polish Journal of Ecology

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2011

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Vol. 59, nr 4

823-828

EN

The assessment of the density and cover of very scarce vegetation in dry habitats may create methodological problems. The variable area transect method (VAT) is a potential labour-saving sampling method and an alternative to plot (quadrate) method. It allows for density estimation without the time-consuming studies associated with other plot-less density estimators. We used the method in a natural shrubland of Saxaul (Haloxylon ammodenderon C.A.M) to define optimum parameters include transect width and individual.s number to which, distance is measured. Three transect widths were chosen, 10-m, 15-m and 20-m and distances to the 3rd, 4th and 5th individual. Transect width affected the estimation, a 20-m width transect had the least relative bias (-0.5%), and a 10-m width sampling had the greatest bias (-20%). However, all methods underestimated the plant density. The most accurate estimation was with the 3rd plant distance and 20-m transect. As the VAT method is more efficient per unit effort in the field than the quadrate methods, it can be recommended for rapid assessment of desert communities density (like saxaul) especially when plants are dispersed at random.

3

Methods for large scale assessment of small mammal abundance in open habitats : plateau pika (Ochotona curzoniae) in alpine grassland

Qu J., Li W., Yang M., Li K., Zhang Y.

Polish Journal of Ecology

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2011

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Vol. 59, nr 4

829-833

EN

Quick and accurate estimation of population density in large scale is required in both scientific studies and wildlife management programs. However, effective estimation of small mammal abundance is usually difficult and timeconsuming due to the body size and wide distribution of these animals. To test the efficiency of different methods in assessing small mammal densities, population dynamics of plateau pikas (Ochotona curzoniae, Hodgson) were studied from April 2005 to August 2009 in alpine grassland (Kobresia humilis) at a height of of 3846 m a.s.l. We compared the precision of walked transects method with mark-recapture method using Efford's maximum likelihood spatial estimator (ML). Significant positive correlation was found between walked transects and Efford's ML estimator (r[^2] = 0.58, P <0.001). The densities calculated with walked transects were about twice lower than those obtained using the mark-recapture method over the study period (i.e., summer). Nevertheless, the walked transects method remains useful for relative density estimation. Hence, the walked transects method is recommended for use as an index of relative density in large-scale assessment in alpine grassland where most small mammals are active and easily detected in an open habitat.