Let (x1,x2,...,xn) be i.i.d. samples from a distribution withmean μ and variance o². Consider the following estimator of µ:лежi=1Σ₁=₁ ixiΣ11(a) Show that µ* is an unbiased estimator of μ.(b) Explain why the above estimator u* is less preferable compared tousing the sample average à as an estimator for μ.Hint: Show that * has a larger mean squared error.Note that xn(n+1)(2n +1)/6.(1/n) (Σï=1 xi), Σ₁_₁ i n(n+1)/2 and Σ₁²i=1

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A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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* Let X1, X2, ..., X, be an independent sam- ple from a normal distribution with unknown mean u and variance o?. Show that the pair (X,5), where X=-Ex,. 5 =(x, - X)'. i=1 is a sufficient statistic for (u, o²). Given A> 0, consider AS as an estimator of o². For what values of A is AS (i) (ii) unbiased? maximum likelihood, Which value of A minimises the mean square error E(A5³ – o*)?
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