Consistent and Asymptotically Normal Estimators

Chapter 3 addresses the concept of consistent and asymptotically normal (CAN) estimators. Suppose $$T_n$$ T n is a consistent estimator of $$\theta $$ θ . In view of the fact that convergence in probability implies convergence in law, $$T_n {\mathop {\rightarrow }\limits ^{L}} \theta , ~~\forall ~~~ \theta $$ T n → L θ , ∀ θ . Thus, the asymptotic distribution of $$T_n$$ T n is degenerate at $$\theta $$ θ . Such a degenerate distribution is not helpful to find the rate of convergence or to find an interval estimator of $$\theta $$ θ . Hence, we try to find a blowing factor $$a_n$$ a n such that the asymptotic distribution of $$a_n ( T_n - \theta )$$ a n ( T n - θ ) is non-degenerate. In particular, we find a sequence $$\{a_n, n \ge 1\}$$ { a n , n ≥ 1 } of positive real numbers tending to $$\infty $$ ∞ as $$n \rightarrow \infty $$ n → ∞ , such that the asymptotic distribution of $$a_n ( T_n - \theta )$$ a n ( T n - θ ) is non-degenerate. It is particularly of interest to find a sequence $$\{a_n, n \ge 1\}$$ { a n , n ≥ 1 } of real numbers tending to $$\infty $$ ∞ as $$n \rightarrow \infty $$ n → ∞ , so that the asymptotic distribution of $$a_n ( T_n - \theta )$$ a n ( T n - θ ) is normal. Estimators for which large sample distribution of $$a_n ( T_n - \theta )$$ a n ( T n - θ ) is normal, are known as CAN estimators. These play a key role in large sample inference theory, in particular, to construct large sample confidence intervals and approximating the distribution of test statistic in large sample test procedures. We discuss variance stabilization technique and studentization technique to construct large sample confidence intervals. In Sects. 3.2 and 3.3 , we investigate various properties of CAN estimators based on sample moments and sample quantiles, for a real as well as a vector parameter respectively and illustrate with several examples. Section 3.4 is concerned with verification of CAN property by simulation using R software.