3. (Two-sided testing) Let X1, ... , Xn be i.i.d. N(u, 1) random variables where u is unknown. (a) Find the likelihood-r

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3 Two Sided Testing Let X1 Xn Be I I D N U 1 Random Variables Where U Is Unknown A Find The Likelihood R 1 (70.51 KiB) Viewed 36 times

3. (Two-sided testing) Let X1, ... , Xn be i.i.d. N(u, 1) random variables where u is unknown. (a) Find the likelihood-ratio test o at significance level a € (0,1) for testing Ho : 1$ [140, H41] against H¡ :E [, M1) such that Ep[0(X)] = E1[0(X)] = a. (It is okay to give a likelihood-ratio test with implicitly defined thresholds because it is not clear how to solve for them. But in theory, this likelihood-ratio test is guaranteed to be uniformly most powerful by Theorem 3.7.1 of the textbook.) (b) Show that there is no uniformly most powerful test at level a for testing Ho : H = against H¡ :fl + Mo. (Hint: Use the part on the uniqueness of the most powerful test in the Neyman-Pearson lemma.)