Question

1. (50 points) Suppose X1, ..., Xn form a random sample from a N(u,02) distribution with p.d.f. Fe 202, for – V2110 <x< . Assume that o = 2 is known. a) (10 points) Derive the 90% confidence interval for u that has the shortest length. You must show all details including the pivot you use. b) (8 points) Show that the sample mean is an efficient estimator for u. Assume in (c)- (f) that the prior distribution of u is a normal distribution with mean 2 and variance 1. c) (10 points) Show that the posterior distribution of u given the data is Normal with mean 8+nX and variance 44 d) (4 points) Find the Bayes estimator w.r.t. to the absolute error loss. Justify your answer.

Answer 1

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P(x) = 1 [n the nth he w? - MZ ~ N( 24 th) d. Let Bayes estimator of the under absolute Loss function is re. Let absolute loss function. Geef) = 1 le-fel .: ELL Gups] = Sheerfe) Plyy life f (e-p) p (lYx de + Sou-ce) P(x) dfe - Leifferentiate who can we ged 112,6@)] = -Species are + S ply) d=0 - Strecerile = Speyote Hence, ce is the posterios median. We know that Mx ~ Normal with posterior mean nt u posterron variance Therefore, Posterior median - Posteriormean Hence, Bayes estimates of pe under absolute loss function. n Je to nãto and nth