Conditions equivalent and doubly equivalent to consistency of approximate MLE's
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Strasser (1981) introduced approximately maximum likelihood estimators (AMLE's) and found a condition equivalent to strong consistency of all AMLE's. Here a condition weaker than that of Strasser is proved to be equivalent to the usual consistency of all AMLE's. Under an additional regularity this condition is shown to be doubly equivalent, which means that it is equivalent to consistency, and its contrary is equivalent to inconsistency of all AMLE's. The doubly equivalent conditions are important—we present an example where MLE is strongly consistent but some AMLE's are inconsistent. It is proved that the additional regularity can be reduced to the finiteness of an observations entropy. All results are motivate and illustrated by examples.
KeywordsMaximum likelihood estimates Approximate maximum likelihood estimates Consistency Inconsistency Observations entropy
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