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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.

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. 2.1 Some examples of estimators Example 1 Let us suppose that {X i}n i=1 are iid normal random variables with mean µ and variance 2.

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