THE UNIVERSITY OF NEW SOUTH WALES DEPARTMENT OF STATISTICS MATH3811/MATH3911- Statistical Inference/Higher Statistical Inference ASSIGNMENT 1 Please, add a cover page containing a copy of your ID card, write with your own handwriting: “I declare that this assignment is my own work, except where acknowledged. I have read and understood the University Rules regarding Academic Misconduct”, and sign it. Assignment due: Thursday, 10th March 2022, 1 pm at the latest. To be submitted via the assignment link on Moodle. Math3811: Attempt all questions except for the star question; Math3911: Attempt all questions. 1. In a manufacturing process in which glass items are being produced, defects in a form of bubbles occur, and if there is more than one bubble, the item is considered defective. The X number of bubbles is thought to follow a Poisson(μ) distribution Pμ(X = x) = f(x, μ) = eμμx x! , x = 0, 1, 2, . . . . Let a sample X = (X1, X2, . . . , Xn) of such Poisson random variables be given. a) Suggest a simple unbiased estimator W of the parameter (μ) = P (X 1) = (1 + μ)eμ (i.e., of the probability of less than equal to one defect). b) Argue that T = Pn i=1Xi is complete and minimal sucient for μ. Hence (or otherwise) derive the UMVUE of (μ). c) What is the MLE of (μ) Find the asymptotic distribution of this MLE (i.e., state the asymptotic distribution of p n( (μ))). d) Defects have been counted on 10 glass items giving the following result 0, 3, 2, 1, 3, 4, 0, 1, 2, 2. Find a point estimate of the probability () using the data given by applying the methods in (b) and (c). Compare the two numerical values and provide a comment. e) Is the variance of the proposed estimator in (b) equal to the variance given by the Cramer-Rao lower bound for an unbiased estimator of () Explain. Calculate the Cramer-Rao lower bound explicitly. f) Find an asymptotic 95% confidence interval for (μ) using the available data and the asymptotic distribution in c). 1 2. Suppose the discrete random X h
as the following probability mass function fX(x; ) = 3 4 ◆x(x+1) 2 1 4 ◆x(x1) 2 x 2 (1 )1×2 , x 2 {1, 0, 1}. a) Is T1 = X sucient for Give reasons. b) Is T2 = X2 sucient for Give reasons. c) Is T1 = X complete for Give reasons. d) Is T2 = X2 complete for Give reasons. 3. Let X1, X2, . . . , Xn be i.i.d. random variables with density f(x; ) = e( x)//, if x 0 else , being unknown parameter and > 0 a fixed constant. a) Sketch a graph of a density from this family for fixed = 1 and = 1. b) Find the cumulative distribution function FX(1)(y; ) and the density fX(1)(y; ) of the smallest of the observations, X(1). (Don’t treat and as specific values any more). Hint: you could use P (X(1) y) = 1 P (X1 y,X2 y, . . . , Xn y) = 1 [P (X1 y)]n. c) Justify that X(1) is a minimal sucient statistic for . d) Find EX(1). e) Show that X(1) is complete. Hence find the UMVUE of . 4. (*) a) i) Suppose we are interested in the parameter and (X1, . . . , Xn) is an esti- mator of with E( ) = b( ). Show that V ar( ) 1 + @b( ) @ 2 IX( ) 1, where IX( ) is the Fisher information. Assume that you can interchange the or- der of integration and di erentiation. You may need to use the well known result (Cauchy-Schwartz Inequality) that for any two random variables U and V we have Cov(U, V )2 Var(U)Var(V ). ii) Suppose X = (X1, X2, …, Xn) are an i.i.d sample from N(0, ) population and we use the MLE = 1 n nX i=1 (Xi Xˉ)2 2 x to estimate the variance . Calculate the bound in (i) and compare it with the variance of the estimator . You may use the fact that (n 1)S2 2n1 where S2 = 1n1 Pn i=1(Xi Xˉ)2. b) i) A random sample of size n is available from the distribution with density f(x; ) = 3x(x+ 1) + 2 e x if x 0, otherwise zero. Find a function of for which there exists an unbiased estimator that attain the Cramer-Rao lower bound. ii) Calculate the Fisher information. iii) Using (a) and (b) or otherwise obtain a closed form expression for Var(Xˉ). 3