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Unformatted text preview: M4056 Lecture Notes. Wednesday, September 29, 2010 MLE for the normal distribution. Suppose vector X = ( X 1 , . . ., X n ) is a sample from a normal distribution with (unknown) parameters and 2 . The maximum likelihood estimates for and 2 are worked out on page 321, by finding the critical points of the two-variable likelihood function. The results are: MLE ( x 1 , . . ., x n ) = x = 1 n n summationdisplay i =1 x i , MLE 2 ( x 1 , . . ., x n ) = 1 n n summationdisplay i =1 ( x i- x ) 2 . It is noteworthy that MLE 2 is not equal to the sample variance: S 2 ( x 1 , . . ., x n ) = 1 n- 1 n summationdisplay i =1 ( x i- x ) 2 . We showed previously that E ( S 2 ) = 2 . Thus, E ( MLE 2 ) = n- 1 n 2 . This shows that MLE 2 is biased it systematically underestimates 2 . Thus, we see a purely mathemat- ical reason for caution in the use of MLEs. They should not be used without evaluating their behavior. We will study this problem in Section 7.3....
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