Term for consistent estimator with respect to $a \to a_0$ not ($n \to \infty$)

Question

When $E[g_n(X)]$ goes to the specific value as $n \to \infty$, it is often said that $g_n$ is a consistent estimator (even if it is not unbiased). Is there any technical term for an estimator $g_{(a)}$ which satisfies the same feature when $a \to a_0$?

Answer 1

Let us assume that $\hat{\theta}_n$ is an unbiased and consistent estimator of $\theta$, and $g$ is continuous a.e. transformation, thus $g(\hat{\theta}_n)$ is consistent estimator of $g(\theta)$ (this theorem called the Continuous mapping theorem). However if $g$ non linear, $g(\hat{\theta}_n)$ will be biased estimator of $g(\theta)$ for every finite $n$. Where the direction of the bias is determined by the curvature of $g$. Namely, if $g$ is convex then $E(g(\hat{\theta}_n) \ge g(E(\hat{\theta}_n))=g(\theta)$ for all $n \in \mathbb{N}$, and vice verse for concave transformations (this called Jensen's inequality).