8 Uniform Estimation Let U1,. . . ,Un i151 Uniform(9, 6) for some unknown 9 6 IR, 9 > 0. We wish to estimate 6 from the data U1,...,U,,. 7 1 '1 (a) Why would using the sample mean U = g E U;- fail in this situation? i=1 (b) Find the PDF of U? for i e {1, . . . ,n}. (c) Consider the following variance estimate: Show that for large n, the distribution of V is close to one of the famous ones, and provide its name and parameters. (d) Use part (c) to construct an unbiased estimator for 0 that uses all the data. (e) Let o2 = Var[U?]. We wish to construct a confidence interval for 02 with a significance level of 8, where 0 < 8 <1. (i) Without any assumption on the magnitude of n, construct a confidence interval for 02 with a significance level of 8 using your estimator from part (d). (ii) Suppose n is large. Construct an approximate confidence interval for 02 with a signifi- cance level of o using your estimator from part (d). You may leave your answer in terms of d and , the normal CDF and its inverse