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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.

Example 2.2.2 (Weibull with known ↵) {Y i} are iid random variables, which follow a Weibull distribution, which has the density ↵y↵1 ↵ exp( ↵(y/ ) ) ,↵>0. Suppose that ↵ is known, but is unknown. Our aim is to fine the MLE of . The log-likelihood is proportional to L n(X; )= Xn i=1 log↵ +(↵ 1)logY i ↵log Y i ↵

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