Transcription of Chapter 2 The Maximum Likelihood Estimator
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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 . In this case we have only one unknown parameter but the minimally sufficient statistics are P P 2.
hood of given (Y i, ... numerical routine may not capture the true maximum) (iii) L n may not be concave, so even if you are close the maximum the numerical routine …
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