Probability and Statistical Models

A statistical model embodies a set of assumptions concerning the generation of the observed data, and similar data from a larger population. A model represents, often in considerably idealized form, the data-generating process. The model assumptions describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. A model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables. As such, "a model is a formal representation of a theory". All statistical hypothesis tests and all statistical estimators are derived from statistical models. More generally, statistical models are part of the foundation of statistical inference. With an emphasis on models and techniques, the book Probability and Statistical Models introduces many of the fundamental concepts of stochastic modeling that are now a vital component of almost every scientific investigation. These models form the basis of well-known parametric lifetime distributions such as exponential, Weibull, and gamma distributions, as well as change-point and mixture models. The book reviews recent developments in theoretical and applied statistical science, highlights current noteworthy results and illustrates their applications; and points out possible new directions to pursue. This book is a must read for probabilists and theoretical and applied statisticians.