Constrained Bayesian Methods of Hypotheses Testing: A New Philosophy of Hypotheses Testing in Parallel and Sequential Experiments

Kartlos Kachiashvili
Tbilisi State University, Tbilisi, Georgia

Series: Mathematics Research Developments
BISAC: MAT027000



Volume 10

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Special issue: Resilience in breaking the cycle of children’s environmental health disparities
Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick


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The problems of one of the basic branches of mathematical statistics – statistical hypotheses testing – are considered in this book. The intensive development of these methods began at the beginning of the last century. The basic results of modern theory of statistical hypotheses testing belong to the cohort of famous statisticians of this period: Fisher, Neyman-Pearson, Jeffreys and Wald (Fisher, 1925; Neyman and Pearson, 1928, 1933; Jeffreys, 1939; Wald, 1947a,b). Many other bright scientists have brought their invaluable contributions to the development of this theory and practice. As a result of their efforts, many brilliant methods for different suppositions about the character of random phenomena are under study, as well as their applications for solving very complicated and diverse modern problems.

Since the mid-1970s, the author of this book has been engaged in the development of the methods of statistical hypotheses testing and their applications for solving practical problems from different spheres of human activity. As a result of this activity, a new approach to the solution of the considered problem has been developed, which was later named the Constrained Bayesian Methods (CBM) of statistical hypotheses testing. Decades were dedicated to the description, investigation and applications of these methods for solving different problems. The results obtained for the current century are collected in seven chapters and three appendices of this book. The short descriptions of existing basic methods of statistical hypotheses testing in relation to different CBM are examined in Chapter One. The formulations and solutions of conventional (unconstrained) and new (constrained) Bayesian problems of hypotheses testing are described in Chapter Two.

The investigation of singularities of hypotheses acceptance regions in CBM and new opportunities in hypotheses testing are presented in Chapter Three. Chapter Four is devoted to the investigations for normal distribution. Sequential analysis approaches developed on the basis of CBM for different kinds of hypotheses are described in Chapter Five. The special software developed by the author for statistical hypotheses testing with CBM (along with other known methods) is described in Chapter Six. The detailed experimental investigation of the statistical hypotheses testing methods developed on the basis of CBM and the results of their comparison with other known methods are given in Chapter Seven. The formalizations of absolutely different problems of human activity such as hypotheses testing problems in the solution – of which the author was engaged in different periods of his life – and some additional information about CBM are given in the appendices.

Finally, it should be noted that, for understanding the materials given in the book, the knowledge of the basics of the probability theory and mathematical statistics is necessary. I think that this book will be useful for undergraduate and postgraduate students in the field of mathematics, mathematical statistics, applied statistics and other subfields for studying the modern methods of statistics and their application in research. It will also be useful for researchers and practitioners in the areas of hypotheses testing, as well as the estimation theory who develop these new methods and apply them to the solutions of different problems.





Chapter 1. Methods of Statistical Hypotheses Testing

Chapter 2. The Bayesian Problem of Hypotheses Testing

Chapter 3. Investigation of Hypotheses Acceptance Regions in Constrained Bayesian Tasks

Chapter 4. Investigations for Normal Distribution

Chapter 5. Sequential Analysis Approach

Chapter 6. Software for Statistical Hypotheses Testing

Chapter 7. Experimental Research

Concluding Remarks


Appendix A

Appendix B

Appendix C

"Hypothesis testing is one of the basic branches of mathematical statistics which is very important for other problems of statistics and has a great application to many theoretical and practical problems. The first statement of the problem and its solution, applying t-test, was realized by Student at the beginning of the previous century." READ MORE... - Alexander Topchishvili, Professor, Dr. Habil. Ing.

"The monograph suggests a new approach to the statistical hypothesis testing, the constrained Bayesian method (CBM). It maintains all the benefits of the basic methods of hypotheses testing. Namely, it uses a data-dependent measure like in Fisher’s test; for making decision it exploits a posteriori probabilities like in Jeffrey’s test and compute Type 1 and Type 2 error probabilities like in Neyman-Pearson’s approach." READ MORE... - Sergei Chobanyan, Visiting Professor of Department of Statistics, Michigan State University, USA

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Keywords: Statistical Hypotheses Testing; Constrained Bayesian Methods; The Parallel Methods; The Sequential Methods; Simple, Composite, Multiple and Directional Hypotheses Testing Methods

The book will be useful for undergraduates and postgraduates in the field of mathematics, mathematical statistics, applied statistics and application of statistical methods in research; Researchers in the areas of hypotheses testing and estimation theory who develop new methods as well as apply these methods to the solution of problems in different spheres of knowledge. The new approach gives to the young generation modern tools for novel, considerable achievements for solving theoretical and applied problems of mathematical statistics. Thus the book is very useful and necessary exactly for the students (coming generation) as it gives them new methods and opportunities for research.

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