Introduction

Chapter 1 is introductory. It discusses basic framework of parametric statistical inference, elaborates on identifiability property of the probability distribution with some illustrations as it is basic to all statistical inference procedures and data analysis. An estimator $$T_n$$ T n is defined as a Borel measurable function from the sample space to the parameter space. In the present book, the focus is on the discussion of large sample optimality properties of estimators and test procedures. Various results from parametric statistical inference for finite sample size, form a foundation of the asymptotic statistical inference theory. Section 1.2 briefly discusses these results. The principal probability tool in asymptotic investigation is the convergence of a sequence of random variables. As sample size increases, we study the limiting behavior of a sequence $$\{T_n, n \ge 1\}$$ { T n , n ≥ 1 } of estimators of $$\theta $$ θ and examine how close it is to $$\theta $$ θ using various modes of convergence. For ready reference, some modes of convergence are defined and various related results are listed in Sect. 1.3. The novelty of the book is use of $$\texttt {R}$$ R software to illustrate various concepts from asymptotic inference. The last section of every chapter is devoted to the application of $$\texttt {R}$$ R software to evaluate the performance of estimators and test procedures by simulation, to obtain solutions of the likelihood equations, to carry out the likelihood ratio test procedures, goodness of fit test procedures and tests for contingency tables. Section 1.4 is devoted to a brief introduction to $$\texttt {R}$$ R , which will be useful to beginners.