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# 4 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT 1302 PROBABILITY AND STATISTICS II (2010-11) EXAMPLE CLASS 4 1. Let X 1 ,X 2 ,X 3 be independent and identically distributed random variables such that E [ X 1 ] = μ (unknown) and Var( X 1 ) = 1 . Two estimators, T 1 and T 2 , have been proposed for estimating μ , defined as follows: T 1 = X 1 + X 2 + X 3 3 and T 2 = X 1 + X 2- X 3 . (a) Show that both T 1 and T 2 are unbiased estimators of μ . (b) It is recommended that T 1 rather than T 2 should be used for estimating μ , even though both estimators are unbiased. Justify this recommendation. [Hint: Compare the MSE.] 2. If X 1 ,X 2 ,...,X n constitute a random sample from the population given by f ( x ) =    e- ( x- δ ) , x > δ , otherwise . (a) Show that ¯ X is a biased estimator of δ . (b) Based on (a), find a unbiased estimator for δ . Is this estimator consistent for δ ?...
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4 - THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS...

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