Tutorial 5

# Tutorial 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2009-10) 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. (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 . 3. Let X 1 ,...,X n be i.i.d. from the Gamma distribution with parameter 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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Tutorial 5 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF...

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