Problem 6. Let Y be Poisson distributed with rate A, that is, P(Y = k) = e AM. "H . Let Y1, . .., Yn ~ Y be an iid sample. (a) Write out the log-likelihood function log C.. (A). (b) Find the MLE An. (c) Find the Fisher information / (1). (d) Plug in A,, to find the estimated SE se(),.). (e) Give the 95% Wald confidence interval for A*. (f) (Computer experiment) Let A* = 2, n = 20. Simulate Yl, ..., Ym ~ Poisson (A*). Perform a Wald test for Ho : A* = 2 at o = 0.05. Repeat 1000 times and count how often you reject the null. How close is the type I error rate to 0.05? (g) Give the 95% Wald confidence interval for * = log (X* ) as the parameter of interest. (h) Repeat the computer experiment in part (f) testing To = log (2) using your confidence interval from (g). (i) Now compare (e) and (g) on the same 1000 repetitions. Which interval is smaller: the log of the interval from (e) or the interval from (g)? How often is the former smaller than the latter? Based on your computer experiments in parts (f) and (h), how do the type I errors compare? What can we conclude if anything