Section 3c

3.3 Unbiased Estimation of sigma^2

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Problem 3c.1

(a)(b)(c)

Suppose that $\mathbf{Y} \sim N_{n}\left(\mathbf{X} \mathbf{\beta }, \sigma^{2} \mathbf{I}_{n}\right)$ , where $\mathbf{X}$ is $n \times p$ of $\operatorname {rank} p$ .

(a)

Find $\operatorname {Var}\left[S^{2}\right]$ .

(b)

Evaluate $\mathbb {E}\left[\left(Y^{T} \mathbf{A}_{1} \mathbf{Y}-\sigma^{2}\right)^{2}\right]$ for

\mathbf{A}_{1} = \frac{1}{n-p+2}\left[\mathbf{I}_{n}-\mathbf{X}\left(\mathbf{X}^{T} \mathbf{X}\right)^{-1} \mathbf{X}^{T}\right]

(c)

Prove that $\mathbf{Y}^{T} \mathbf{A}_{1} \mathbf{Y}$ is an estimate of $\sigma^{2}$ with a smaller meansquared error than $S^{2}$ .

Problem 3c.2

Let $Y_{1}, Y_{2}, \ldots , Y_{n}$ be independently and identically distributed with mean $\theta$ and variance $\sigma^{2}$ . Find the nonnegative quadratic unbiased estimate of $\sigma^{2}$ with the minimum variance.