Question

Consider X1,X2, , Xn be an iid random sample fron Unif(0.0). Let θ = (끄+1) Y where Y = max(X1, x. . . . , X.). It can be easily shown that the cdf of Y is h(y) = Prp.SH-()" 1. Prove that Y is a biased estimator of θ and write down the expression of the bias 2. Prove that θ is an unbiased estimator of θ. 3. Determine and write down the cdf of 0 4. Discuss why this expression of F(0) is a good candidate for pivotal quantity and indicate its distribution a > b1/m to construct 5. Consider a confidence level 1-α, then use the facts that am < b the 100(1-a)% upper confidence bound for θ, namely Ca (X1, . . . , Xn)-UB,-a(X) such that PrWBi-a(X) > 이 = 1-0. Hint You must draw your region under the curve of the distribution of your pivotal quantity, and then algebraically manipulate the expression to obtain UB(x) 6, Compute the actual 95% upper confidence bound for θ with the data set 2.42,5.35, 4.80, 6.95,0.76,5.66, 2.88, 0.78, 6.54 7. Consider testing the hypotheses Ho :02 11 vs Ha 0<11 1. Unsing the fact that F(0)Unif(0,1), write down a plausible test statistic when Ho is true, and compute its observed value for the above data 2. Compute the Pvalue, namely, pvalue-Pr(9 < θ[H0 is true] 3. Draw the rejection region for this test at0.15 (Hint: standard uniform), and place the above computed value of the test statistic on the curve. 4. What do you conclude from this test?

Answer 1

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