Introduction

The initial descriptive statistics of the collected data (graphical analysis, data summaries, etc.) is often useful for understanding its main features and selecting an appropriate statistical model, which is a key part of the second stage. Incorrect modeling of the data makes its further analysis and inference meaningless. This chapter starts from likelihood function that plays a fundamental role in statistical theory. To verify completeness for a general distribution can be a nontrivial mathematical problem. The verification of completeness is much simpler for the exponential family of distributions that includes many of the “common” distributions. Completeness is often used to prove the uniqueness of various estimators. Completeness of a sufficient statistic yields its minimal sufficiency. The chapter introduces an important family of distributions that include many “standard” distributions (e.g., exponential, binomial, Poisson, and normal) and discusses its main common properties.