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# 05 - THE UNIVERSITY OF HONG KONG Department of Statistics...

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THE UNIVERSITY OF HONG KONG Department of Statistics And Actuarial Science STAT 1302 PROBABILITY AND STATISTICS II EXAMPLE CLASS 5 1. Let X 1 , ..., X n be independent Poisson random variables with X j having parameter , where λ > 0 is an unknown parameter. (a) Find the Fisher information contained in ( X 1 , ..., X n ) about λ . (b) Find the MLE of λ . What is its i. Bias; ii. Variance; iii. Mean squared error? iv. Is this MLE of λ efficient? 2. Let X 1 , ..., X n be i.i.d. from the uniform distribution over the interval [ θ, θ +1], where θ is unknown. (a) Find a bivariate sufficient statistic for θ . (b) Find a maximum likelihood estimator of θ . (c) Show that max { X 1 , ..., X n } - n 1+ n is an unbiased estimator of θ . 1

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3. (a) The likelihood function of λ is l x ( λ ) = p ( x | λ ) = n Y j =1 P ( ) = n Y j =1 ( ) x j exp {- } x j ! = Q n j =1 j x j Q n j =1 x j ! · λ n j =1 x j · exp {- n (1 + n ) 2 · λ } C ( x ) λ n j =1 x j exp {- n (1 + n ) 2 · λ } , The log-likelihood function is then S x ( λ ) = log C ( x ) + log λ · n X j =1 x j - n (1 + n ) 2 · λ.
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05 - THE UNIVERSITY OF HONG KONG Department of Statistics...

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