Quantile-based reliability analysis
This book introduces quantile-based reliability analysis. It gives a novel approach to reliability theory using quantile functions in contrast to the traditional approach based on distribution functions. Even though quantile functions and distribution functions are mathematically equivalent ways to define a probability distribution, the book shows that quantile functions have several advantages over distribution functions: (1) many data sets with non-elementary distribution functions can be modeled by quantile functions with simple forms; (2) most quantile functions approximate many of the standard models in reliability analysis quite well, and hence an arbitrary quantile function will be a good first approximation if physical conditions do not suggest a plausible model; (3) the inference procedures for quantile models need less information and are more robust to outliers. By the way, the book is written or as said in its preface, the book is biased towards the mathematical theory, with examples intended to clarify various notions and applications to real data being limited to demonstrate the utility of quantile functions. This book has nine chapters. Chapter 1 deals with the definition, properties and various descriptive measures based on the quantile functions including order statistics and \(L\)-moments, and demonstrates some graphic methods. Chapter 2 discusses various reliability concepts like hazard rate, and mean residual life, in the conventional form as well as their quantile equivalents. Chapter 3 gives a detailed presentation of the distributional and reliability aspects of quantile function models along with some applications to real data. Chapter 4 describes different aging concepts in quantile versions. Chapter 5 covers total time on test transforms, an essentially quantile-based notion in details. Chapter 6 discusses the \(L\)-moments and partial moments in relation to residual life along with their use in inferential methods, which are alternatives to the conventional moments. Chapter 7 introduces bathtub hazard models along with their quantile counterparts and some new quantile functions that exhibit nonmonotone hazard quantile functions. In Chapter 8, the definitions and properties of various stochastic orders encountered in reliability theory are described. Finally, Chapter 9 deals with various methods of estimation and modeling problems. This book has a broad applicability across fields such as statistics, survival analysis, economics, engineering, demography, insurance, and medical science. It can be used as an excellent reference book for faculty and professionals. Although the book does not provide exercises, it gives a clear presentation with many examples, figures, and tables. Thus, it can serve as a textbook for advanced undergraduate and graduate courses in reliability and statistics.
scientific article; zbMATH DE number 6176764