Midterm I S - 1 (a). The likelihood function based on X = (...

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Unformatted text preview: 1 (a). The likelihood function based on X = ( X 1 ,X 2 ,...,X n ) is given by L ( X 1 ,X 2 ,...,X n ; , ) = n exp {- n i =1 X i } n productdisplay i =1 X i ( ) = C ( ) , exp { Q ( ) n i =1 X i } h ( X ) We can see that it constitutes an exponential family of distributions with parameter . Since Q ( ) =- is a decreasing function in , it follows from the general theory for exponential families that the UMP test for H : versus H 1 : > reject the alternative hypothesis when n i =1 X i < C , for some critical value C which has to chosen such that P ( n i =1 X i < C ) = . (b). The critical value C is determined by the following equation P ( n i =1 < C ) = . In order to be able to apply the CLT, we first standardize the test statistic n i =1 X i . With the help of the hint we get E n i =1 X i = n E( X i ) = n and Var( n i =1 X i ) = n Var( X i ) = n 2 . Thus we obtain P parenleftBig...
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This note was uploaded on 04/20/2008 for the course STATS 131C taught by Professor Polonik during the Spring '07 term at UC Davis.

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Midterm I S - 1 (a). The likelihood function based on X = (...

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