Consistency of an Estimator

As discussed in Chap. 1 , in asymptotic inference theory, 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. The most frequently investigated large sample property of an estimator is weak consistency. Weak consistency of an estimator is defined in terms of convergence in probability. We examine how close the estimator is to the true parameter value in terms of probability of proximity. Weak consistency is always referred to as consistency in literature. In Sect. 2.1, we define it for a real parameter and illustrate by a variety of examples. We study some properties of consistent estimators, the most important being the invariance of consistency under continuous transformation. Strong consistency and uniform consistency of an estimator are discussed briefly in Sects. 2.3 and 2.4. In Sect. 2.5, we define consistency when the distribution of a random variable or a random vector is indexed by a vector parameter. It is defined in two ways as marginal consistency and joint consistency, the two approaches are shown to be equivalent. This result is heavily used in applications. Thus, to obtain a consistent estimator for a vector parameter, one can proceed marginally and use all the tools discussed in Sect. 2.2. From examples in Sects. 2.2 and 2.5, we note that, for a given parameter, one can have an uncountable family of consistent estimators and hence one has to deal with the problem of selecting the best from the family. It is discussed in Sect. 2.6. Within a family of consistent estimators of $$\theta $$ θ , the performance of a consistent estimator is judged by the rate of convergence of a true coverage probability to 1 and of MSE to 0 for a consistent estimator whose MSE exists, faster the rate better is the estimator. Section 2.7 is devoted to the verification of the consistency of an estimator by simulation. It is illustrated through some examples and $$\texttt {R}$$ R software.