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For over 200 years, practitioners have been developing parametric families of probability distributions for data analysis. More recently, an active development of nonparametric and semiparametric families has occurred. This book includes an extensive discussion of a wide variety of distribution families—nonparametric, semiparametric and parametric—some well known and some not. An all-encompassing view is taken for the purpose of identifying relationships, origins and structures of the various families. A unified methodological approach for the introduction of parameters into families is developed, and the properties that the parameters imbue a distribution are clarified. These results provide essential tools for intelligent choice of models for data analysis. Many of the results given are new and have not previously appeared in print. This book provides a comprehensive reference for anyone working with nonnegative data.