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Chapter 2 The Maximum Likelihood Estimator

Chapter 2. The Maximum Likelihood Estimator We start this Chapter with a few quirky examples , based on estimators we are already familiar with and then we consider classical Maximum Likelihood estimation. Some examples of estimators Example 1. Let us suppose that {Xi }ni=1 are iid normal random variables with mean and variance 2 . P. The best estimators unbiased estimators of the mean and variance are X = n1 ni=1 Xi P P P 2. and s2 = n 1 1 ni=1 (Xi X )2 respectively. To see why recall that i X i and i Xi P P 2. are the sufficient statistics of the normal distribution and that i Xi and i Xi are complete minimal sufficient statistics. Therefore, since X and s2 are functions of these minimally sufficient statistics, by the Lehmann-Sche e Lemma, these estimators have minimal variance. Now let us consider the situation where the mean is and the variance is 2.

butions from the which the parameter comes from is known, then the maximum likelihood 56. estimator of the parameter ,whichisdefinedas b ...

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