Question 3 [15 marks] Let X1, ... , X10 be an i.i.d. sample from the uniform distribution on (0 - 1,0+1). With U = max{X

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Question 3 15 Marks Let X1 X10 Be An I I D Sample From The Uniform Distribution On 0 1 0 1 With U Max X 1 (40.12 KiB) Viewed 27 times

Question 3 [15 marks] Let X1, ... , X10 be an i.i.d. sample from the uniform distribution on (0 - 1,0+1). With U = max{X1,...,X10} and V = min{X1,..., X10}, any value between U - 1 and V + 1 is an mle for 0. In particular, consider Ô - (U – 1) + (V +1) U + V 2 2 which has the density function = fo(x) = 5{x - (0-1)}', 0-15050, 5{(@+1) – x}, 0<x<0+1. (1) Find the density function and cdf of ô - . (ii) Using (i), find the shortest exact 90% confidence interval for 6. Simplify the numerical expressions but do not compute their values. Give a reason why your confidence interval is the shortest. (iii) Suppose you use this confidence interval to test H, : 0 = 0 versus H : 0 +0. For what values of 0 does this test have power equal to 1? Simplify the numerical expressions but do not compute their values.