Parameter Estimation and Evaluation Maximum Likelihood Estimation Maximum Likelihood Estimation Example 4(8.4) Suppose X"is an IID N(u,o2)random sample.Find the MLE for(4,o2). Solution .Put 0=(u,o2).Then the log-likelihood function of Xm hiaX9y=-he)-分ao2-a】 (X:-)2 2=1 To be Continued Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 31/207
Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 31/207 Maximum Likelihood Estimation Maximum Likelihood Estimation Example 4 (8.4) Solution To be Continued
Parameter Estimation and Evaluation Maximum Likelihood Estimation Maximum Likelihood Estimation ●The FOC: alni(X") ∂μ x-四=0 i=1 alni(X") 2a∑(X:-2=0, 2 m 0g2 i=1 where 0 =(2,62).Note that here,we treat o2 instead of o as a parameter. ●It follows that A=Xn; 2=n1(X:-Xn)2. i=1 To be Continued Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 32/207
Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 32/207 Maximum Likelihood Estimation Maximum Likelihood Estimation To be Continued
Parameter Estimation and Evaluation Maximum Likelihood Estimation Maximum Likelihood Estimation The MLE 62 for o2 differs slightly from the sample vari- ance S%=(n-1)-1∑1(X,-Xn)2. ●1 To check the SOC,we compute the sample Hessian ma- trix 02 To be Continued Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 33/207
Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 33/207 Maximum Likelihood Estimation Maximum Likelihood Estimation To be Continued
Parameter Estimation and Evaluation Maximum Likelihood Estimation Maximum Likelihood Estimation 。When 0=0,we have which is negative definite.Hence,0 is a local maximizer. Note that the MLE =(A,62)is a sufficient statistic for 0=(,02). Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 34/207
Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 34/207 Maximum Likelihood Estimation Maximum Likelihood Estimation
Parameter Estimation and Evaluation Maximum Likelihood Estimation Maximum Likelihood Estimation Question: What is the value of In L(X")when 0=0=(d,62)? Suppose we treat o instead of o2 as a parameter.Can we obtain the same MLE solution in the above example? The answer is yes.We will obtain 6=V62. This follows from the invariance property of MLE. Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21,2020 35/207
Parameter Estimation and Evaluation Parameter Estimation and Evaluation Introduction to Statistics and Econometrics April 21, 2020 35/207 Maximum Likelihood Estimation Maximum Likelihood Estimation