Wyniki wyszukiwania - BazTech

1

Asymptotics of Monte Carlo maximum likelihood estimators

Miasojedow B., Niemiro W., Palczewski J., Rejchel W.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 2

295--310

EN

We describe Monte Carlo approximation to the maximum likelihood estimator in models with intractable norming constants and explanatory variables. We consider both sources of randomness (due to the initial sample and to Monte Carlo simulations) and prove asymptotical normality of the estimator.

2

Professor Ryszard Zieliński's contribution to Monte Carlo methods and random number generators. Uniform asymptotics in statistics

Niemiro W.

Mathematica Applicanda

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2012

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Vol. 40, No. 2

92--106

PL

The aim of the paper is to summarize contributions of Ryszard Zieliński to two important areas of research. First, we discuss his work related to Monte Carlo methods. Ryszard Zieliński was particularly interested in Monte Carlo optimization. About 10 of his papers concerned stochastic algorithms for seeking extrema. He examined methods related to stochastic approximation, random search and global optimization. We stress that Zielinski often considered computational problems from a statistical perspective. In several articles he explicitly indicated that optimization can be reformulated as a statistical estimation problem. We also discuss relation between the family of Simulated Annealing algorithms on the one hand and some procedures examined earlier by Ryszard Zieliński on the other. Another topic belonging to Monte Carlo methods, in which Ryszard Zieliński has achieved interesting results, is construction of random number generators and examination of their statistical properties. Zieliński proposed an aperiodic generator based on Weil sequences and showed how it can be efficiently implemented. Later he constructed an algorithm which uses several such generators and produces pseudo-random sequences with better statistical properties. The second area of Zieliński’s work discussed here is related to uniform limit theorems of mathematical statistics. We stress the methodological motivation behind the research in this direction. In Zieliński’s view, asymptotic results should hold uniformly with respect to the family of probability distributions under consideration. In his opinion, this requirement comes from the very nature of statistical models and the needs of practical applications. Zieliński examined uniform versions the Weak Law of Large Numbers, Strong Law of Large Numbers and Central Limit Theorem in several statistical models. Some results were rather unexpected. He also gave a necessary and sufficient condition for uniform consistency of sample quantiles. Two papers of Ryszard Zieliński were devoted to uniform consistency of smoothed versions of empirical cumulative distribution function. In one of them he proved a version of Dvoretzky-Kiefer-Wolfowitz inequality. The aim of the paper is to summarize contributions of Ryszard Zieliński to two important areas of research. First, we discuss his work related to Monte Carlo methods. Ryszard Zieliński was particularly interested in Monte Carlo optimization. About 10 of his papers concerned stochastic algorithms for seeking extrema. He examined methods related to stochastic approximation, random search and global optimization. We stress that Zielinski often considered computational problems from a statistical perspective. In several articles he explicitly indicated that optimization can be reformulated as a statistical estimation problem. We also discuss relation between the family of Simulated Annealing algorithms on the one hand and some procedures examined earlier by Ryszard Zieliński on the other. Another topic belonging to Monte Carlo methods, in which Ryszard Zieliński has achieved interesting results, is construction of random number generators and examination of their statistical properties. Zieliński proposed an aperiodic generator based on Weil sequences and showed how it can be efficiently implemented. Later he constructed an algorithm which uses several such generators and produces pseudo-random sequences with better statistical properties. The second area of Zieliński’s work discussed here is related to uniform limit theorems of mathematical statistics. We stress the methodological motivation behind the research in this direction. In Zieliński’s view, asymptotic results should hold uniformly with respect to the family of probability distributions under consideration. In his opinion, this requirement comes from the very nature of statistical models and the needs of practical applications. Zieliński examined uniform versions the Weak Law of Large Numbers, Strong Law of Large Numbers and Central Limit Theorem in several statistical models. Some results were rather unexpected. He also gave a necessary and sufficient condition for uniform consistency of sample quantiles. Two papers of Ryszard Zieliński were devoted to uniform consistency of smoothed versions of empirical cumulative distribution function. In one of them he proved a version of Dvoretzky-Kiefer-Wolfowitz inequality.

3

Remarks on uniform convergence of random variables and statistics

Niemiro W., Zieliński R.

Demonstratio Mathematica

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2012

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Vol. 45, nr 2

265-274

EN

Convergence in distribution, convergece in probability, and convergence almost surely, uniform with respect to a family of probability distributions is considered. These concepts appeared to be appropriate tools for asymptotic theory of mathematical statistics and many partial results are scattered in the literature of the subject. The aim of this note is to present a unified review of the results in a general and abstract setup. We examine a few rather paradoxical examples which hopefully shed some light on the subtleties of the underlying definitions and the role of asymptotic approximations in statistics. A motivation for considering these problems is provided by their applications.

4

A new method for identifying outlying subsets of data

Zalewska M., Grzanka A., Niemiro W., Samoliński B.

Control and Cybernetics

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2008

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Vol. 37, no 3

693-709

EN

In various branches of science, e.g. medicine, economics, sociology, it is necessary to identify or detect outlying subsets of data. Suppose that the set of data is partitioned into many relatively small subsets and we have some reason to suspect that one or several of these subsets may be atypical or aberrant. We propose applying a new measure of separability, based on the ideas borrowed from the discriminant analysis. In our paper we define two versions of this measure, both using a jacknife, leave-one-out, estimator of classification error. If a suspected subset is significantly well separated from the main bulk of data, then we regard it as outlying. The usefulness of our algorithm is illustrated on a set of medical data collected in a large survey "Epidemiology of Allergic Diseases in Poland" (ECAP). We also tested our method on artificial data sets and on the classical IRIS data set. For a comparison, we report the results of a homogeneity test of Bartoszyński, Pearl and Lawrence, applied to the same data sets.

5

Positron emission tomography by Markov chain Monte Carlo with auxiliary variables : a basic algorithm

Lasota S., Niemiro W., Koronacki J.

Prace Instytutu Podstaw Informatyki Polskiej Akademii Nauk

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1999

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nr 884

3-26

EN

In the report, an algorithm for positron emission tomography (PET) image reconstruction is proposed. The algorithm belongs to the family of Markov chain Monte Carlo methods with auxiliary variables. The well-known model of Vardi et al. (1985) is used for PET. The fact that an image consists of finitely many, in fact relatively few, gray-levels of unknown values is explicity used to advantage: in the algorithm, the levels are represented by a fixed number of labels, so that at one step of the algorithm current approximation to the image is easily described by a configuration of finitely many labels and at another step real-valued intensities are assigned to each label. The algorithm decomposes naturally into the image restoration algorithm and the additional reconstruction (of generalized deconvolution) step. Simulation results are included which suggest that the method proposed is truly reliable and worth further study leading to practical implementation.

PL

W raporcie przedstawiony jest nowy algorytm rekonstrukcji obrazów uzyskiwanych w tomografii pozytronowej (positron emission tomography, PET). Algorytm składa się z części służącej do oczyszczania poissonowsko zaszumionych obrazów opisywanych znaną liczbą intensywności Poissona oraz z kroku służącego do rekonstrukcji (uogólnionej dekonwulcji) obrazu. Zaproponowany algorytm nalezy do rodziny metod Monte-Carlo typu łańcuchów Markowa (Markov chain Monte Carlo) ze zmiennymi pomocniczymi typu Swendsena-Wanga oraz "rozprzęganiem" podobnym do zaproponowanego przez Higdona. Zadanie PET rozwiązane jest dla znanego modelu Vardiego i in. (1985). Zawarte w raporcie wyniki badań symulacyjnych pozwalają algorytm uznać za zdecydowanie zasługujący na opracowanie jego praktycznej implementacji. (W obecnej postaci algorytm jest wiarygodny ale zbyt wolny; jego szybka wersja będzie przedmiotem oddzielnego opracowania).