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Hi
please help me this one
For data that can be modeled with normal distributions, our basic model takes the form Y ∼ N(Xβ, V ), where Y is the vector of all the responses from all the subjects. Throughout this question, assume that X has full column rank and that both V and XT V −1X are non-singular. Look at the class of estimators of β given by
β ̃(W) = XT WX−1 XT WY , where W is any N ×N matrix for which XTWX is non-singular. These estimators are linear functions of the data vector Y .
(a) Show that all the estimators in the above class are unbiased for β.
(b) The best estimator in this class, known as the BLUE (best linear unbiased estimator) is the one with W = V −1, which has vari- ance XT V −1X−1. It is best in the following sense: suppose you want to estimate the function tT β for any vector t using a linear estimator of the data λTt Y that is unbiased. Note that this is an estimator that is linear in the data and depends on the vector t. Here unbiased means that E λTt Y = tT β, regardless of the value of β or the choice of t. Show that the condition of unbiasedness implies that λt must satisfy XT λt = t.
(c) Now show that, among all the linear unbiased estimators, the minimum variance is achieved by the estimator tT βˆ, which is β ̃(W ) computed with W = V −1.
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