Characterization Problems In Mathematical Statistics

This book deals with the characterization of probability distributions. It is well known that both the sum and the difference of two Gaussian independent random variables with equal variance are independent as well. The converse statement was proved independently by M. Kac and S. N. Bernstein. This result is a famous example of a characterization theorem. In general, characterization problems in mathematical statistics are statements in which the description of possible distributions of random variables follows from properties of some functions in these variables. In recent years, a great deal of attention has been focused upon generalizing the classical characterization theorems to random variables with values in various algebraic structures such as locally compact Abelian groups, Lie groups, quantum groups, or symmetric spaces. The present book is aimed at the generalization of some well-known characterization theorems to the case of independent random variables taking values in a locally compact Abelian group $X$. The main attention is paid to the characterization of the Gaussian and the idempotent distribution (group analogs of the Kac-Bernstein, Skitovich-Darmois, and Heyde theorems). The solution of the corresponding problems is reduced to the solution of some functional equations in the class of continuous positive definite functions defined on the character group of $X$. Group analogs of the Cramer and Marcinkiewicz theorems are also studied. The author is an expert in algebraic probability theory. His comprehensive and self-contained monograph is addressed to mathematicians working in probability theory on algebraic structures, abstract harmonic analysis, and functional equations. The book concludes with comments and unsolved problems that provide further stimulation for future research in the theory.

The problem of identifiability is basic to all statistical methods and data analysis, occurring in such diverse areas as Reliability Theory, Survival Analysis, and Econometrics, where stochastic modeling is widely used. Mathematics dealing with identifiability per se is closely related to the so-called branch of "characterization problems" in Probability Theory. This book brings together relevant material on identifiability as it occurs in these diverse fields.

Provides the necessary skills to solve problems in mathematical statistics through theory, concrete examples, and exercises With a clear and detailed approach to the fundamentals of statistical theory, Examples and Problems in Mathematical Statistics uniquely bridges the gap between theory andapplication and presents numerous problem-solving examples that illustrate the relatednotations and proven results. Written by an established authority in probability and mathematical statistics, each chapter begins with a theoretical presentation to introduce both the topic and the important results in an effort to aid in overall comprehension. Examples are then provided, followed by problems, and finally, solutions to some of the earlier problems. In addition, Examples and Problems in Mathematical Statistics features: Over 160 practical and interesting real-world examples from a variety of fields including engineering, mathematics, and statistics to help readers become proficient in theoretical problem solving More than 430 unique exercises with select solutions Key statistical inference topics, such as probability theory, statistical distributions, sufficient statistics, information in samples, testing statistical hypotheses, statistical estimation, confidence and tolerance intervals, large sample theory, and Bayesian analysis Recommended for graduate-level courses in probability and statistical inference, Examples and Problems in Mathematical Statistics is also an ideal reference for applied statisticians and researchers.

Problems of calculating the reliability of instruments and systems and the development of measures to increase efficiency and reduce operational costs confronted physicists and mathe maticians at the end of the '40's and the beginning of the '50's in connection with the unrelia bility of electro-vacuum instruments used in aviation. Since then steadily increasing demands for the accuracy, reliability and complexity required in electronic equipment have served as a stimulus in the development of the theory of reliability. From 1950 to 1955 Epstein and Sobel [67,68] and Davis [62], in an analysis of statistical data of the operating time of an instrument up to failure, showed that the distribution is exponential in many cases. Consequently, the ex ponential distribution became basic to research associated with experiments on life expectancy. Further research has shown that there are a whole series of problems in reliability theory for which the exponential distribution is inapplicable. However, it can practically always be used as a first approximation. The ease of computational work due to the nice properties of the exponential distribution (for example, the lack of memory property, see Section 1) is also a reason for its frequent use. AB a rule, data on the behavior of the failure rate function are used to test the hypothesis that a given distribution belongs to the class of exponential distributions, and order statistics are used to estimate the parameter of the exponential distribution.

Problems of calculating the reliability of instruments and systems and the development of measures to increase efficiency and reduce operational costs confronted physicists and mathe maticians at the end of the '40's and the beginning of the '50's in connection with the unrelia bility of electro-vacuum instruments used in aviation. Since then steadily increasing demands for the accuracy, reliability and complexity required in electronic equipment have served as a stimulus in the development of the theory of reliability. From 1950 to 1955 Epstein and Sobel [67,68] and Davis [62], in an analysis of statistical data of the operating time of an instrument up to failure, showed that the distribution is exponential in many cases. Consequently, the ex ponential distribution became basic to research associated with experiments on life expectancy. Further research has shown that there are a whole series of problems in reliability theory for which the exponential distribution is inapplicable. However, it can practically always be used as a first approximation. The ease of computational work due to the nice properties of the exponential distribution (for example, the lack of memory property, see Section 1) is also a reason for its frequent use. AB a rule, data on the behavior of the failure rate function are used to test the hypothesis that a given distribution belongs to the class of exponential distributions, and order statistics are used to estimate the parameter of the exponential distribution.