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**Unformatted text preview: **MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 19 Konrad Menzel April 28, 2009 Maximum Likelihood Estimation: Further Examples Example 1 Suppose X N ( , 2 ) , and we want to estimate the parameters and 2 from an i.i.d. sample X 1 , . . . , X n . The likelihood function is n 1 ( X i ) 2 L ( ) = e 2 2 2 i =1 It turns out that its much easier to maximize the log-likelihood, n ( X i ) 2 log L ( ) = log 2 1 e 2 2 i =1 n = log 1 ( X i ) 2 2 2 2 i =1 n = n 2 log(2 2 ) 1 ( X i ) 2 2 2 i =1 In order to find the maximum, we take the derivatives with respect to and 2 , and set them equal to zero: n n 1 1 0 = 2 2 i =1 2( X i ) = n i =1 X i Similarly, n n n n 2 1 1 1 0 = 2 2 2 + 2 2 2 i =1 ( X i ) 2 2 = n i =1 ( X i ) 2 = n i =1 ( X i X n ) 2 Recall that we already showed that this estimator is not unbiased for 2 , so in general, Maximum Likelihood Estimators need not be unbiased. Example 2 Going back to the example with the uniform distribution, suppose X U [0 , ] , and we are interested in estimating . For the method of moments estimator, you can see that 1 ( ) = E [ X ] = 2 1 so equating this with the sample mean, we obtain MoM = 2 X n What is the maximum likelihood estimator? Clearly, we wouldnt pick any max { X 1 , . . . , X n } because a sample with realizations greater than has zero probability under . Formally, the likelihood is 1 n if 0 X i for all i = 1 , . . . , n L ( ) = 0 otherwise We can see that any value of max { X 1 , . . . , X n } cant be a maximum because L ( ) is zero for all those points. Also, for max { X 1 , . . . , X n } the likelihood function is strictly decreasing in , and therefore, it is maximized at MLE = max { X 1 , . . . , X n } Note that since X i < with probability 1, the Maximum Likelihood estimator is also going to be less than with probability one, so its not unbiased. More specifically, the p.d.f. of X ( n ) is given by n 1 y n 1 f X ( n ) ( y ) = n [ F X ( y )] n 1 f X ( y ) = if 0 y 0 otherwise so that y n n E [ X ( n ) ] = yf X ( n ) ( y ) dy = n n + 1 dy = We could easily construct an unbiased estimator = n +1 X ( n ) . n 1.1 Properties of the MLE The following is just a summary of main theoretical results on MLE (wont do proofs for now) If there is an ecient estimator in the class of consistent estimators, MLE will produce it....

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