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Middle East Technical University Faculty of Economics and Administrative Sciences Department of Economics Econ 206 – Fall, 2014 November 21 PROBLEM SET 4 1. Let Y1, Y2, . . . , Yn denote a random sample from a population with mean μ and variance σ2. Consider the following three estimators for μ: a. Show that each of the three estimators is unbiased. b. Find the efficiency of ̂ μ3 relative to ̂ μ2 and ̂ μ1 respectively. Solution: 2. Suppose that Y1, Y2, . . . , Yn is a random sample from a normal distribution with mean μ and variance σ2. Two unbiased estimators of σ2 are ̂1 2 relative to σ ̂2 2 Find the efficiency of σ Solution: 1 3. Let Y1 , Y2 , …….,Yn denote a random sample from the probability density function θ-1 f(y)= { θy ,0<y<1 0 ,elsewhere θ where θ>0. Show that ̅ Y is a consistent estimator for θ+1 Solution: 2 4. Suppose that populations are normally distributed with σ21 = σ22 = σ2 . Show that 2 ̅ ) + ∑n (Yi -Y ̅) ∑ni=1 (Xi -X i=1 2n-2 2 is a consistent estimator of σ2 . Solution: 5. If X1 , X2 , …….,Xn constitute a random sample of size n from an exponential population, show that ̅ X is a sufficient estimator of the parameter θ. Solution: 3 6. Let Y1, Y2, . . . , Yn denote independent and identically distributed random variables from a. Power family distribution with parameters α and θ (α, θ > 0) If θ is known, show that ∏ni=1 Yi is sufficient for α. b. Pareto distribution with parameters α and β (α, β > 0) If β is known, show that ∏ni=1 Yi is sufficient for α. Solution: 4