STAT2006 Assignment 2 Deadline 6 March 2022 Question 1. (25 marks) Let {1,2, … ,} be an independent random sample from Poisson () , where is an unknown parameter. Suppose that the sample size is large. (a) Show that both the sample mean and the sample variance 2 are unbiased estimator of . (b) Calculate the Cramer-Rao lower bound of unbiased estimators of . (c) Which one of the estimators in part (a) should be preferred and why (d) What is the large sample size distribution of (e) Using part (d), or otherwise, derive an approximate 1 confidence interval for . Hint: 2 < < 2 ≈ 1 leads to ( )( ) < 0 ≈ 1 , where < are the roots of the quadratic equation 2 2 + 22 + 2 = 0. Question 2. (25 marks) Let {1,2, … ,} be an independent random sample from Exponential(), where is an unknown parameter. Consider the method of moment estimator = 1 . (a) Using moment generating function or otherwise, show that ~Gamma(,) (b) Use the result of part (a), show that is a biased estimator of . Hint: 1 ∞ 0 = Γ() (c) Show that = ( + 2)2( 1)( 2), where = + 2. Hint: Evaluate 2 and, hence, = 22( 1)2( 2). (d) Is a consistent estimator of Why Hint: Show that lim →∞ = 0. (e) Use the result of part (a), or otherwise, find an unbiased estimator of . (f) Show that is also the maximum likelihood estimator of . Show that the large sample size distribution of is , 2 . (g) Show that an approximate 95% confidence interval for is 1 + 1.96 √ , 1 1.96 √ . Hint: 1.96 < √ < 1.96 = 0.95 Question 3 (Difficult question). (15 marks) Let {1,2, … ,} be a random sample from the uniform distribution on [, 2]. Suppose that is a large integer. Let (1) = min 1≤≤ and () = max 1≤≤ . (a) Show that the joint pdf of (1),() is (1) = ,() = = 1 ( 1)( ) 2, where < ≤ < 2. Hence find the marginal pdf of (1) and () respectively. Hint: For < ≤ < 2, show that (1) < ,() < = () < (1) ≥ ,() < = . The joint pdf of (1),() is (1) = ,() = = 2 (1) < ,() < . (b) Show that +1 +2 (1) is an unbiased estimator of . Hint: Show that the marginal distribution of (1) is (1) = = (2 ) 1, < < 2. Then, find (1) . (c) Show that the maximum likelihood estimator of is = 1 2 (). However, is not an unbiased estimator of . Hint: Show that the likelih
ood function is a decreasing function of . Hence, the value of that maximizes the likelihood function is at the minimum value of . To show that is not an unbiased estimator of , find the marginal distribution of () and then show that () = 2+1+1 . (d) Show that = +1 2+1 () is an unbiased estimator of . (e) (Optional) Show that + 1 + 2(1) > 1, for large value of . Which unbiased estimator is preferred Why Question 4. (20 marks) Let {1,2, … ,} be a random sample from a population with pdf ( = = 2 2 , > 00, otherwise, where is an unknown parameter. (a) Show that the moment estimator (MME) of is = 4 2 . Hint: Use ∫ 2 √22 ∞ ∞ 2 22 = 2, show that () = √ 2 . (b) Show that the maximum likelihood estimator (MLE) of is = 1 2 =1 . (c) Is the MLE above (i) consistent (ii) unbiased Give reasons. Hint: Show that 2~Exponential 1 . (d) Derive the Cramer-Rao lower bound for the unbiased estimators of and comment on the MLE. Question 5. (15 marks) A certain stimulus is to be tested for its effect on blood pressure. Twelve men have their blood pressure measured before and after the stimulus. The results are Man 1 2 3 4 5 6 7 8 9 10 11 12 Before () 120 124 130 118 140 128 140 135 126 130 126 127 After () 128 131 131 127 132 125 141 137 118 132 129 135 In this particular scenario, we define = as the improvement or increase in blood pressure due to the stimulus and assume that ’s ( = 1,2, … ,12) are independent and distributed according to a (,2 ) distribution. Let () = and () = . (a) Construct a 95% confidence interval for , the average effect of the stimulus on blood pressure. Do you think the stimulus does have an effect on the blood pressure based on this 95% confidence interval Briefly explain your answer. Note that ~ , 2 . (b) Construct an equal tailed 95% confidence interval for . Hint: Use ( 1)2 2 ~ 12 . 1 2, 12 < ( 1)22 < 2, 12 = 1 Question 6. (20 marks) Let {1,2, … ,} be a random sample from ( ,2) , and {1,2, … ,} be a random sample from (,2). Assume that the two random samples are independent. Denote = ( ) ( ) 2 + 2 and = ( 1)2 2 + ( 1)22 . (a) If and are independent, 2 2 = , and is a known constant, construct a random variable which has a distribution. Hint: Show that = ( 1)2 2 + ( 1)22 ~+ 22 (b) Using , construct a 1 confidence interval for . Hint: 2 ,+ 2 < < 2 ,+ 2 = 1 .