STA261H1 Study Guide - Independent And Identically Distributed Random Variables, Bias Of An Estimator, Linear Combination
13 Feb 2014
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3ny3n 1 (cid:82) : a) the mle isy(n) (see solution for problem 8 in homework 4) 4 = y , so = 4 y. 3: the pdf of y(n) is fy(n)(y) = nf (y)n 1f (y) = n( y3. Hence, the mle is not unbiased but (3n+1)y(n) n 1 not n 1. 3n n: a) the rst factor should be n , as some of you noticed, to get an unbiased estimator. The estimator i wrote is not unbiased but it"s asymptotically unbiased. Below is the proof with the corrected factor in front. is an unbiased estimator of . The proof consists in using the lln twice. (even the estimator with n 1. Consistent estimators are about large sample (as n goes to in nity) but the second estimator is the sample mean of only two observations, so it doesn"t really make sense to talk about consistency for this estimator.
