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Unformatted text preview: Middle East Technical University Faculty of Economics and Administrative Sciences Department of Economics Econ 206 – Spring 2008 PROBLEM SET # 3 1. Let 1 X and 2 X be a random sample of two observations from a population with mean μ and variance 2 σ . Consider the following three point estimators of μ : (1) (2) 1 2 1 2 1 2 1 1 1 3 1 2 ˆ ˆ 2 2 4 4 3 3 X X X X X X X μ μ = + = + = + a. Which of these estimators is the most efficient? b. Find the relative efficiency of X with respect to each of the other two estimators. 2. Let 1 2 , ,..., n X X X denote a random sample of size n from a Poisson distribution with mean θ and variance θ . a. Find the Cramer Rao Lower Bound (CRLB) for θ . b. Show that the sample mean is an efficient estimator. c. Let 1 7 ˆ 3 n X X X θ + + = is an estimator of θ . Derive the efficiency of the sample mean relative to ˆ θ . 1 2 n ( ) __ If X ,X ,...,X constitute a random sample from the population given by; for x f(x) 0 otherwise Show that X is a biased estimator of...
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