Understanding asymptotic consistency and its unique advantages in large sample statistical inference

The main objective of this paper is to investigate the usefulness of asymptotic consistency in large-sample statistical inference. In many statistical applications, plug-in methods are used to construct statistics, which raises the natural question of whether the asymptotic properties of $T_n({\hat{\beta }}_n,{\hat{\alpha }}_n)$ remain unchanged when the estimator ${\hat{\alpha }}_n$ is replaced by its corresponding target quantity ${\alpha }_n$. We establish that if ${\hat{\alpha }}_n$ is asymptotically consistent for ${\alpha }_n$, meaning that ${lim}_{n\to \infty }P({\hat{\alpha }}_n={\alpha }_n)=1$, then $T_n({\hat{\beta }}_n,{\hat{\alpha }}_n)$ and $T_n({\hat{\beta }}_n,{\alpha }_n)$ share identical asymptotic behaviors in terms of convergence and limiting distribution. This result notably simplifies the derivation of asymptotic properties, especially when the dependency between ${\hat{\beta }}_n$ and ${\hat{\alpha }}_n$ is complex. Furthermore, we systematically explore the relationship between asymptotic consistency and traditional forms of consistency, such as weak and strong consistency, clarifying their distinctions through theorems and counterexamples. Finally, the theoretical findings are demonstrated via three specific applications, illustrating the practical benefits of asymptotic consistency in large-sample inference.