Maximum Likelihood Estimate

Suppose that a value is such that

for every .

Then is the maximum likelihood estimate (MLE) of .

The MLE can be computed for every dataset , so the function

is called maximum likelihood estimator of .

Invariance of the MLE[edit | edit source]

If is a MLE for the original parametrization and if is a 1-1 function defined on ,

then

is a MLE in the new parametrization.

PROOF

Asymptotic Properties of MLE[edit | edit source]

THEORY 1

Under some regularity conditions on the statistical model {}

  • the MLE of ,, exists almost surely
  • as

THEORY 2

Under some regularity conditions on the statistical model {}

as

where

{} Fisher information

{} observed Fisher information

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