Fuzzy probability and statistics.

*(English)*Zbl 1095.62002
Studies in Fuzziness and Soft Computing 196. Berlin: Springer (ISBN 3-540-30841-5/hbk). xiii, 270 p. (2006).

This book is written in the following divisions: (1) the introductory chapters consisting of Chapters 1 and 2; (2) introduction to fuzzy probability in Chapters 3–5; (3) introduction to fuzzy estimation in Chapters 6–11; (4) estimators of probability density functions based on a fuzzy maximum entropy principle in Chapters 12–14; (5) introduction to fuzzy hypothesis testing in Chapters 15–18; (6) fuzzy correlation and regression in Chapters 19–25; (7) fuzzy ANOVA model in Chapters 26 and 27; (8) a fuzzy estimator of the median in nonparametric statistics in Chapter 28; and (9) random fuzzy numbers with applications to Monte Carlo studies in Chapter 29.

The beginning introduction to fuzzy sets, fuzzy logic and relations on fuzzy number domains is given in Chapter 2, which is based on D. Walton, Fallacies arising from ambiguity. (1996; Zbl 0895.03002). Introduction to fuzzy probability in Chapters 3–5 is based on J. J. Buckley, Fuzzy probabilities. New approach and applications. (2003; Zbl 1017.60001). Some new algorithms are presented here. Introduction to fuzzy estimation. (Chapters 6–11) is based on J. J. Buckley, Fuzzy statistics. (2004; Zbl 1076.62029). Fuzzy estimators for arrival and services rates are, respectively, from J. J. Buckley, Fuzzy probabilities and fuzzy sets for web planning. (2004; Zbl 1032.60002), and J. J. Buckley, Simulating fuzzy systems. (2005; Zbl 1098.93019). Also, fuzzy estimators for uniform probabilities can be found here, but the derivation of these estimators is new.

The fuzzy probability density estimators based on a fuzzy maximum entropy principle are obtained in Chapter 12. In the Chapters 13 and 14 crisp probability densities are given. Introduction to fuzzy hypothesis testing in the following Chapters is based on the author’s book, “Fuzzy statistics.” (op.cit.). The Chapters 19–25 on fuzzy correlation and regression also come from this book. The results on fuzzy ANOVA (Chapters 26 and 27), a fuzzy estimator for the median (Chapter 28) and fuzzy random numbers (Chapter 29) have not been published before. Chapter 30 contains selected Maple/Solver commands used in this book to solve optimization problems or to generate the figures. The final chapter has a summary and suggestions for future research. Resuming, the reviewed book is an interesting and well organized monograph on fuzzy statistics which can be recommended for specialists and non-specialists in the field of fuzzy sets research.

The beginning introduction to fuzzy sets, fuzzy logic and relations on fuzzy number domains is given in Chapter 2, which is based on D. Walton, Fallacies arising from ambiguity. (1996; Zbl 0895.03002). Introduction to fuzzy probability in Chapters 3–5 is based on J. J. Buckley, Fuzzy probabilities. New approach and applications. (2003; Zbl 1017.60001). Some new algorithms are presented here. Introduction to fuzzy estimation. (Chapters 6–11) is based on J. J. Buckley, Fuzzy statistics. (2004; Zbl 1076.62029). Fuzzy estimators for arrival and services rates are, respectively, from J. J. Buckley, Fuzzy probabilities and fuzzy sets for web planning. (2004; Zbl 1032.60002), and J. J. Buckley, Simulating fuzzy systems. (2005; Zbl 1098.93019). Also, fuzzy estimators for uniform probabilities can be found here, but the derivation of these estimators is new.

The fuzzy probability density estimators based on a fuzzy maximum entropy principle are obtained in Chapter 12. In the Chapters 13 and 14 crisp probability densities are given. Introduction to fuzzy hypothesis testing in the following Chapters is based on the author’s book, “Fuzzy statistics.” (op.cit.). The Chapters 19–25 on fuzzy correlation and regression also come from this book. The results on fuzzy ANOVA (Chapters 26 and 27), a fuzzy estimator for the median (Chapter 28) and fuzzy random numbers (Chapter 29) have not been published before. Chapter 30 contains selected Maple/Solver commands used in this book to solve optimization problems or to generate the figures. The final chapter has a summary and suggestions for future research. Resuming, the reviewed book is an interesting and well organized monograph on fuzzy statistics which can be recommended for specialists and non-specialists in the field of fuzzy sets research.

Reviewer: Krzysztof Piasecki (Poznań)

##### MSC:

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

03E72 | Theory of fuzzy sets, etc. |