PRACTICE MID SESSION TEST MATH5905 Time allowed: 135 minutes 1. Let X = (X1, X2, . . . , Xn) be i.i.d. Poisson(θ) random variables with density function f(x, θ) = e−θθx x! , x = 0, 1, 2, . . . , and θ > 0. a) The statistic T (X) = ∑n i=1Xi is complete and sufficient for θ. Provide justifi- cation for why this statement is true. b) Derive the UMVUE of h(θ) = e−kθ where k = 1, 2, . . . , n is a known integer. You must justify each step in your answer. Hint: Use the interpretation that P (X1 = 0) = e −θ and therefore P (X1 = 0, . . . , Xk = 0) = P (X1 = 0)k = e−kθ. c) Calculate the Cramer-Rao lower bound for the minimal variance of an unbiased estimator of h(θ) = e−kθ. d) Show that there does not exist an integer k for which the variance of the UMVUE of h(θ) attains this bound. e) Determine the MLE hˆ of h(θ). f) Suppose that n = 5, T = 10 and k = 1 compute the numerical values of the UMVUE in part (b) and the MLE in part (e). Comment on these values. g) Consider testing H0 : θ ≤ 2 versus H1 : θ > 2 with a 0-1 loss in Bayesian setting with the prior τ(θ) = 4θ2e−2θ. What is your decision when n = 5 and T = 10. You may use: ∫ 2 0 x12e−7xdx = 0.00317 Note: The continuous random variable X has a gamma density f with param- eters α > 0 and β > 0 if f(x;α, β) = 1 Γ(α)βα xα−1e−x/β and Γ(α + 1) = αΓ(α) = α! 1 2. Let X1, X2, . . . , Xn be independent random variables, with a density f(x; θ) = { e−(x−θ), x > θ, 0 else where θ ∈ R1 is an unknown parameter. Let T = min{X1, . . . , Xn} = X(1) be the minimal of the n observations. a) Show that T is a sufficient statistic for the parameter θ. b) Show that the density of T is fT (t) = { ne−n(x−θ), t > θ, 0 else Hint: You may find the CDF first by using P (X(1) < x) = 1− P (X1 > x ∩X2 > x · · · ∩Xn > x). c) Find the maximum likelihood estimator of θ and provide justification. d) Show that the MLE is a biased estimator. Hint: You might want to consider using a substitution and then utilize the density of an exponential distribution when computing the integral. e) Show that T = X(1) is complete for θ. f) Hence determine the UMVUE of θ.
