What are the appropriate smoothness conditions that will ensure a maximum likelihood estimator is consistent?

Question

According to a theorem, under appropriate smoothness conditions on $f$ , the maximum likelihood estimator from an i.i.d. sample is consistent. So what are the conditions that will ensure that the estimators are consistent?

Answer 1

Yes, is some places the conditions are not specified clearly. One set of sufficien requisites are Wald constrains:

1. Different models have different parameter values
2. The set of parameter allowable values is compact
3. The likelihood function is continuous.
4. The set of probable samples (with density>0) does not depends on the parameter value.

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