derive the maximum likelihood estimator for b when a is known, i.e. only b is a statistical parameter.

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Question 1.Derive the expressions for the method of moments estimates for a and b. Compute and report the estimates of a and b for the given data.

Question 2. Use the optim function to calculate numerically and report the maximum likelihood estimate of θ = (a, b) for the given data.

HINT: I suggest that you use the following function to compute the log-likelihood. ll.gamma <- function(theta, xs) { return(sum(dgamma(xs, shape=theta[1], rate=theta[2], log=TRUE))) }

Question 3.  Approximate the mean-squared error of the Method of Moments and Maximum Likelihood estimators of each component of θ for θ = (a, b) = (2, 1.5) when n = 100.

Question 4. Derive the Maximum Likelihood estimator for b when a is known, i.e. only b is a statistical parameter.

Question 5. a. Compute an observed 95% Wald confidence interval for b given the data, assuming it is known that a = 2. b. Compute an approximation of the coverage of the corresponding confidence interval when (a, b) = (2, 1.5) and n = 100. 3 Testing

Question 6.  Derive the generalized likelihood ratio (GLR) test statistic for the null hypothesis that a = 2 (with no restriction on b), and state its asymptotic distribution. 1

Question 7  Report and interpret the p-value for the hypothesis test corresponding to Question 6, for the data provided.

Question 8 Consider a significance level of α = 0.05, and assume that b = 1.5 and n = 100. Compute an approximation of the Type I error rate of the hypothesis test described in Question 6, and explain why it is not exactly equal to α.

Question 9 Consider a significance level of α = 0.05, and assume that b = 1.5 and n = 100. Approximate the power of the hypothesis test described in Question 6 when a = 2.5, and explain the source of any approximation error.

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