Asymptotic Theory

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

On the Chebyshev-Cramér Asymptotic Expansions

Summary The mixed linear model is a popular method for analysing unbalanced repeated measurement data. The classical statistical tests for parameters in this model are based on asymptotic theory that is unreliable in the small samples that are often encountered in practice. For testing a given fixed effect parameter with a small sample, we develop and investigate refined likelihood ratio (LR) tests. The refinements considered are the Bartlett correction and use of the Cox–Reid adjusted likelihood; these are examined separately and in combination. We illustrate the various LR tests on an actual data set and compare them in two simulation studies. The conventional LR test yields type I error rates that are higher than nominal. The adjusted LR test yields rates that are lower than nominal, with absolute accuracy similar to that of the conventional LR test in the first simulation study and better in the second. The Bartlett correction substantially improves the accuracy of the type I error rates with either the conventional or the adjusted LR test. In many cases, error rates that are very close to nominal are achieved with the refined methods.

Previous article Next article On the Extension of the Class of Stable DistributionsV. M. KruglovV. M. Kruglovhttps://doi.org/10.1137/1117081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. V. Gnedenko and , A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass., 1962 Google Scholar[2] Yu. V. Linnik, Decomposition of probability distributions, Edited by S. J. Taylor, Dover Publications Inc., New York, 1964xii+242 MR0189085 (32:6512) 0121.35502 Google Scholar[3] B. V. Gnedenko, Some theorems on distribution functions, Učenie Zapiski Moskovskogo Univ., 30 (1939), 61–82, (In Russian.) Google Scholar[4] V. M. Kruglov, A note on infinitely divisible distributions, Theory Prob. Applications, 15 (1970), 319–324 10.1137/1115038 0329.60011 LinkGoogle Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions31 October 2021 | Journal of the Indian Society for Probability and Statistics, Vol. 22, No. 2 Cross Ref Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable lawStatistics & Probability Letters, Vol. 137 Cross Ref A Log Log Law for Subsequences of Delayed Random Sums27 July 2017 | Journal of the Indian Society for Probability and Statistics, Vol. 18, No. 2 Cross Ref Merging to Semistable ProcessesI. 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A Non-Uniform Estimate for the Convergence Speed in the Multi-Dimensional Central Theorem

The goal of this communication is to present a weighted likelihood discriminant for minimum error shape classification. Different from traditional maximum likelihood (ML) methods in which classification is carried out based on probabilities from independent individual class models as is the case for general hidden Markov model (HMM) methods, our proposed method utilizes information from all classes to minimize classification error. Proposed approach uses a hidden Markov model as a curvature feature based 2D shape descriptor. In this contribution we present a generalized probabilistic descent (GPD) method to weight the curvature likelihoods to achieve a discriminant function with minimum classification error. In contrast with other approaches, a weighted likelihood discriminant function is introduced. We believe that our sound theory based implementation reduces classification error by combining hidden Markov model with generalized probabilistic descent theory. We show comparative results obtained with our approach and classic maximum-likelihood calculation for fighter planes in terms of classification accuracies

This book presents the theory of probability and mathematical statistics at a level suitable for researchers at the frontiers of applied disciplines. Examples and exercises make essential concepts in measure theory and analysis accessible to those with preparation limited to vector calculus. Complete, detailed solutions to all the exercises demonstrate techniques of problem solving and provide immediate feedback. Part I, The Theory of Probability, starts with elementary set theory and proceeds through basic measure and probability, random variables, integration and mathematical expectation. It concludes with an extensive survey of models for distributions of random variables. Part II, The Theory of Statistics, begins with sampling theory and distribution theory for statistics from normal populations, proceeds to asymptotic (large-sample) theory, and on to point and interval estimation and tests of parametric hypotheses. The last three chapters cover tests of nonparametric hypotheses, Bayesian methods, and linear and nonlinear regression. Researchers and graduate students in applied fields such as actuarial science, biostatistics, economics, finance, mathematical psychology, and systems engineering will find this book to be a valuable learning tool and an essential reference. Contents: The Theory of Probability: Probability on Abstract Sets Probability on Sets of Real Numbers Mathematical Expectation Models for Distributions The Theory of Statistics: Sampling Distributions Asymptotic Distribution Theory Estimation Tests of Parametric Hypotheses Tests of Nonparametric Hypotheses A Survey of Bayesian Methods Regression Analysis Readership: Researchers and graduate students in applied fields such as actuarial science, biostatistics, economics, finance, mathematical psychology, and systems engineering. Key Features: Present basic concepts in real and complex analysis and measure theory along with the applications to probability Difficult theoretical points are followed directly by examples and exercises with detailed solutions Provide a survey of Bayesian methods and an extensive treatment of goodness-of-fit tests Suitable learning tool for students and researchers in applied fields for whom a rigorous foundation in probability and statistical inference is essential

Decision-making processes in medicine rely increasingly on modelling and simulation techniques; they are especially useful when combining evidence from multiple sources. Markov models are frequently used to synthesize the available evidence for such simulation studies, by describing disease and treatment progress, as well as associated factors such as the treatment's effects on a patient's life and the costs to society. When the same decision problem is investigated by multiple stakeholders, differing modelling assumptions are often applied, making synthesis and interpretation of the results difficult. This paper proposes a standardized approach towards the creation of Markov models. It introduces the notion of ‘general Markov models’, providing a common definition of the Markov models that underlie many similar decision problems, and develops a language for their specification. We demonstrate the application of this language by developing a general Markov model for adverse event analysis in radiotherapy and argue that the proposed method can automate the creation of Markov models from existing data. The approach has the potential to support the radiotherapy community in conducting systematic analyses involving predictive modelling of existing and upcoming radiotherapy data. We expect it to facilitate the application of modelling techniques in medical decision problems beyond the field of radiotherapy, and to improve the comparability of their results.

Accounting for uncertainty in control-relevant statistics

In this paper, we address the problem of reduced-complexity estimation of general large-scale hidden Markov models with underlying nearly completely decomposable discrete-time Markov chains and finite-state outputs. An algorithm is presented that computes O(e) (where e is the related weak coupling parameter) approximations to the aggregate and full-order filtered estimates with substantial computational savings. These savings are shown to be quite large when the chains have blocks with small individual dimensions. Some simulation studies are presented to demonstrate the performance of the algorithm.

Leroux's method for general hidden Markov models

This paper characterizes the asymptotic behaviour of the likelihood ratio test statistic (LRTS) for testing homogeneity (i.e. no mixture) against gamma mixture alternatives. Under the null hypothesis, the LRTS is shown to be asymptotically equivalent to the square of Davies's Gaussian process test statistic and diverges at a log n rate to infinity in probability. Based on the asymptotic analysis, we propose and demonstrate a computationally efficient method to simulate the null distributions of the LRTS for small to moderate sample sizes.

The book Turbulent Combustion by Norbert Peters is a concise monograph on single-phase gaseous low Mach number turbulent combustion. It is compiled from the author's review papers on this topic plus some additional material. Norbert Peters characterizes turbulent combustion both by the way fuel and air are mixed and by the ratio of turbulent and chemical time scales. This approach leads naturally to detailed models, which are based on results of turbulence modelling and asymptotic flame theory. In both areas Norbert Peters has contributed significantly over the last two decades. The book has four sections. In chapter 1 he discusses briefly the state of the art of combustion models as they are used by different authors. Important turbulent and chemical scales are introduced, which are then used to introduce and explain the different combustion models. He distinguishes between premixed and non-premixed combustion and also between infinitely fast and finite rate chemistry. The current turbulent combustion models are described in order of their complexity and physical accuracy. He explains Eddy BreakUp and Eddy Dissipation Models, the fundamentals of the PDF transport equation, and the laminar flamelet concept applied to non-premixed and premixed turbulent combustion. Then, the Conditional Moment Closure, the Linear Eddy Model, and combustion models used in Large Eddy Simulations are described very briefly. Chapter 2 is devoted to premixed turbulent combustion. After introducing some characteristic dimensionless numbers Peters uses the level set approach and the flamelet concept to formulate a combustion model valid in the thin zone and corrugated reaction zone regimes. He shows parallels between this more fundamental model and standard models like the Bray-Moss-Libby model. He also presents models for the turbulent burning velocity, the Flame Surface Area Ratio, and discusses the effects of gas expansion. Very helpful for the reader's understanding is the presentation of three worked examples of a slot burner, a propagating spherical flame and an oscillating counter flow. Peters' model for premixed turbulent combustion is based on the equations for the mean and the variance of the $G$-equation, some closure relations as well as the flamelet equation for premixed combustion. A numerical example is used to discuss its accuracy. Non-premixed turbulent combustion is the subject matter of chapter 3. Peters uses the mixture fraction variable and asymptotic flame theory to explain the regimes of non-premixed turbulent combustion. Two worked examples of a counterflow diffusion flame and the one-dimensional unsteady laminar mixing layer help the reader to understand the theory. After discussing turbulent jet diffusion flames and introducing the flamelet equation he develops steady and unsteady flamelet models for non-premixed turbulent combustion. In particular, the Eulerian Particle Flamelet Model and the RIF (Representative Interactive Flamelet) Model are discussed. These models have been used to predict pollutant formation in a gas turbine and a direct injection Diesel engine, respectively. Finally, partially premixed combustion is discussed in chapter 4. Lifted turbulent diffusion flames are reviewed and the prediction of the lift-off height is identified as a key problem. This leads directly to the introduction of the concept of a triple flame. Different models for partially premixed combustion are then presented and the numerical simulation and the scaling of lift-off heights in turbulent jet flames are studied. There is no doubt that this book is well written and is an important contribution to combustion literature. Peters uses asymptotic theory and scale separation to develop combustion models from first principles. Also, the book contains a comprehensive review of the current literature on turbulent combustion. It is clearly a `must have' for experienced combustion modellers and experimentalists. The book has not been written for non-experts and beginners; these readers would probably have liked more worked examples and exercises, a list of symbols, some material on the mathematical techniques of asymptotic analysis, and a more detailed discussion of the standard combustion models that are often used in the literature. Also, the numerical aspects of turbulent combustion modelling are not discussed in the book. Nevertheless, the book will be a useful source for advanced courses on combustion. Markus Kraft

This article evaluates the accuracy of approximate confidence intervals for parameters and quantiles of the smallest extreme value and normal distributions. The findings also apply to the Weibull and the lognormal distributions. The interval estimates are based on (a) the asymptotic normality of the maximum likelihood estimator, (b) the asymptotic x 2 distribution of the likelihood ratio (LR) statistic, (c) a mean and variance correction to the signed square roots of the LR statistic, and (d) the Bartlett correction to the LR statistic. The extensive Monte Carlo results about true error probabilities and average lengths under various degrees of censoring show advantages of the LR-based intervals. For complete or moderately censored samples, the mean and variance correction to the LR statistic gives nearly exact and symmetric error probabilities. In small samples with heavy censoring, the Bartlett correction tends to give conservative error probabilities, whereas the uncorrected LR interval is often anticonservative. The results also indicate that LR-based methods have longer interval lengths than intervals based on the asymptotic normality of the maximum likelihood estimator.

CHAPTER ONE - KUBEIAGENESIS