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Consider regression model y = XB + €, where X is an n X p matrix with rank(X) p; (0,022), and > is known n X n positive definite matrix Please find an u...

Question

Consider regression model y = XB + €, where X is an n X p matrix with rank(X) p; (0,022), and > is known n X n positive definite matrix Please find an unbiased estimator of 02 and prove it_

Consider regression model y = XB + €, where X is an n X p matrix with rank(X) p; (0,022), and > is known n X n positive definite matrix Please find an unbiased estimator of 02 and prove it_



Answers

Given multivariate data $\mathbf{X}_{1}, \ldots, \mathbf{X}_{N}$ (in $\mathbb{R}^{p} )$ in meandeviation form, let $P$ be a $p \times p$ matrix, and define $\mathbf{Y}_{k}=P^{T} \mathbf{X}_{k}$ for $k=1, \ldots, N$
a. Show that $\mathbf{Y}_{1}, \ldots, \mathbf{Y}_{N}$ are in mean-deviation form. $[\text { Hint: }$ Let $\mathbf{w}$ be the vector in $\mathbb{R}^{N}$ with a 1 in each entry. Then $\left[\begin{array}{lll}{\mathbf{x}_{1}} & {\cdots} & {\mathbf{X}_{N}}\end{array}\right] \mathbf{w}=\mathbf{0}\left(\text { the zero vector in } \mathbb{R}^{p}\right) \cdot ]$
b. Show that if the covariance matrix of $\mathbf{X}_{1}, \ldots, \mathbf{X}_{N}$ is $S$ then the covariance matrix of $\mathbf{Y}_{1}, \ldots, \mathbf{Y}_{N}$ is $P^{T} S P .$


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