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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 ; λ,β ) = λ nβ exp { λ Σ n i =1 X i } n productdisplay i =1 X β i Γ( β ) = C ( λ ) , exp { Q ( λ )Σ n i =1 X i } h ( X ) We can see that it constitutes an exponential family of distributions with parameter λ . Since Q ( λ ) = λ is a decreasing function in λ , it follows from the general theory for exponential families that the UMP test for H : λ ≥ λ versus H 1 : λ > λ reject the alternative hypothesis when Σ n i =1 X i < C α , for some critical value C α which has to chosen such that P λ (Σ n i =1 X i < C α ) = α . (b). The critical value C α is determined by the following equation P λ (Σ n i =1 < C α ) = α . 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 EΣ n i =1 X i = n E( X i ) = n β λ and Var( ∑ n i =1 X i ) = n Var( X i ) = n β λ 2 . Thus we obtain P λ parenleftBig...
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 Spring '07
 Polonik
 Statistics, Derivative, Normal Distribution, exponential family, Σn Xi

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