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I am studying statistical inference. I want to know what should be the general strategy for proving the consistency of an estimator.

In most problems whenever I prove consistency, I usually see whether the estimator is a function of the maximum likelihood estimator (MLE). I have seen in Casella and Berger's text that MLE estimator are consistent in most cases. And also, functions of MLE estimators are also MLE. I just wanted to know whether my approach to proving the consistency is right. If not, can you suggest some ways by which I can approach proofs of consistency?

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I think there are a number of approaches. Other than MLE-related approaches, two that I happen to have used are:

  • Consistency is preserved under a continuous transformation. See Casella-Berger pp. 233 Theorem 5.5.4.
  • Asymptotic normality implies consistency. See Casella-Berger pp. 472-473 Example 10.1.13.

I'd be interested to hear other approaches from the community!

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