NotesNov19 - Maximum likelihood estimator may or may not be...

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Maximum likelihood estimator may or may not be unbiased. The bias, if present, be- comes small as the sample size increases. Maximum likelihood estimators are consis- tent, and for large sample sizes their MSE is the smallest possible. If ˆ θ is the MLE of an unknown parameter θ , and h is a one-to-one function, then h ( ˆ θ ) is a MLE of h ( θ ).
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Estimation by the method of moments The moments of the distribution of the sample (the population moments) depend on the unknown parameters. One can estimate these moments from the sample, resulting in the sample moments. Equate populations moments and the sam- ple moments, and solve the resulting sys- tem of equations for the unknown param- eters The solution is a moment estimator of the parameters.
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Sample moments : Given a sample X 1 , . . . , X n , the k th sample mo- ment is ˆ m k = 1 n n X i =1 X k i , k = 1 , 2 , . . . . The first sample moment = the sample mean: ˆ m 1 = ¯ X n .
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The procedure : Choose the number k of the moments to
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