Let X1,..., X. Uniform(0,6). Let X(1) and X(n) be the X smallest and largest order statistics of X1, ..., Xn. (i) Derive

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Let X1 X Uniform 0 6 Let X 1 And X N Be The X Smallest And Largest Order Statistics Of X1 Xn I Derive 1 (71.35 KiB) Viewed 67 times

Let X1,..., X. Uniform(0,6). Let X(1) and X(n) be the X smallest and largest order statistics of X1, ..., Xn. (i) Derive the joint distribution of X(1) and X(n)- (ii) Find the MLE (Ô) for parameter 8. What are the bias and MSE of ô? Show you exact calculations. (iii) Suppose we want to test H. : 01 << Qu vs. Hų: < 0 or 6 > Qu, where OL and Ou are known values. A reasonable test statistic is the MLE Ô. A reasonable rejection region is when ő is either too small or too big, i.e., reject H, when ên < C or ôn > Cu. The upper threshold can be set to Cu = ou. Find the lower threshold Cų for a level a test. As the null hypothesis is composite, please note that Cu should satisfy supoeloz,Qu] PÔ <Cluôn > Cu) Sa. (iv) Find the power function (0) = Pocê <Cuôn > Cu). Hint: you need to discuss 7(@) for < CL, CL < a < Cu, and e > Cu. =