# Strategies for proving that an estimator is consistent

When starting out in theoretical statistics, it’s often difficult to know how to even get started on the problem at hand. It helps to have a few general strategies in mind. In this post, I list some strategies you can use to prove that an estimator $\hat{\theta}_n$ is consistent for a parameter $\theta$.

Note:

1. This list is not exhaustive. If you have a strategy for proving consistency that is not in the list, feel free to share it!)
2. The arguments here are not necessarily rigorous, so you need to check the conditions under which they apply.

Now for the strategies:

1. Use the (weak) law of large numbers. This is especially applicable for i.i.d. sums.
2. Use Chebyshev’s inequality: If $\hat\theta_n$ is unbiased, then $\mathbb{P} \{ |\hat\theta_n - \theta | > \epsilon \} \leq \dfrac{Var(\hat\theta_n)}{\epsilon^2}$. Thus, the estimator will be consistent if $Var(\hat\theta_n) \rightarrow 0$.
3. Actually in strategy 2, we only need $\hat\theta_n$ to be asymptotically unbiased (i.e. $\mathbb{E}[\hat\theta_n] \rightarrow \theta$) and the result would still hold.
4. If your estimator is the maximum likelihood estimator (MLE), then it is consistent under some regularity conditions (see here).
5. If your estimator is an M-estimator or a Z-estimator, try to use the argmax consistency theorem (e.g. see slides 7 and 9 here).
6. Try to use the continuous mapping theorem: if $X_n \stackrel{P}{\rightarrow} X$, then $g(X_n) \stackrel{P}{\rightarrow} g(X)$ for any continuous function $g$.