5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 5 1. Let X 1 ,...,X n be independent Poisson random variables with X j having pa- rameter , where λ > 0 is an unknown parameter. Given that the Fisher information contained in ( X 1 ,...,X n ) about λ is n ( n +1) 2 λ . 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 . 3. Let X 1 ,...,X n be i.i.d. from the Gamma distribution with parameters α and β , which f ( x ) = 1 Γ( α ) β α x α - 1 e - x/β , obtain the method-of-moments estimators for α and β . 1
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This note was uploaded on 04/20/2011 for the course STAT 1302 taught by Professor Smslee during the Spring '10 term at HKU.

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5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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