Math Problems Vault

Problem 1b.1

Suppose that $X_{1}, X_{2}$ , and $X_{3}$ are random variables with common mean $\mu$ and variance matrix

\operatorname {Var}\left[\mathbf{X}\right] = \sigma ^{2}\left[\begin{smallmatrix} 1 & 0 & 0 \\ 0 & 1 & \frac{1}{4} \\ 0 & \frac{1}{4} & 1 \end{smallmatrix}\right]

Find $\mathbb {E}\left[X_{1}^{2}+2 X_{1} X_{2}-4 X_{2} X_{3}+X_{3}^{2}\right]$ .

?

Problem 1b.2

If $X_{1}, X_{2}, \ldots , X_{n}$ are independent random variables with common mean $\mu$ and variances $\sigma_{1}^{2}, \sigma_{2}^{2}, \ldots , \sigma_{n}^{2}$ , prove that $\sum_{i}\left(X_{i}-\bar{X}\right)^{2} /[n(n-1)]$ is an unbiased estimate of $\operatorname {Var}\left[\bar{X}\right]$ .

?

Problem 1b.3

(a)(b)

Suppose that in Exercise 2 the variances are known. Let $\bar{X}_{w} = \sum_{i} w_{i} X_{i}$ be an unbiased estimate of $\mu$ (i.e., $\sum_{i} w_{i} = 1$ ).

?

(a)

Prove that $\operatorname {Var}\left[\bar{X}_{w}\right]$ is minimized when $w_{i} \propto 1 / \sigma_{i}^{2}$ . Find this minimum variance $v_{\text{min }}$ .

(b)

Let $S_{w}^{2} = \sum_{i} w_{i}\left(X_{i}-\bar{X}_{w}\right)^{2} /(n-1)$ . If $w_{i} \sigma_{i}^{2} = a(i = 1,2, \ldots , n)$ , prove that $\mathbb {E}\left[S_{w}^{2}\right]$ is an unbiased estimate of $v_{\text{min }}$ .

Problem 1b.4

(a)(b)

The random variables $X_{1}, X_{2}, \ldots , X_{n}$ have a common nonzero mean $\mu$ , a common variance $\sigma^{2}$ , and the correlation between any pair of random variables is $\rho$ .

?

(a)

Find $\operatorname {Var}\left[\bar{X}\right]$ and hence prove that $-1 /(n-1) \leq \rho \leq 1$ .

(b)

If

Q = a \sum _{i = 1}^{n} X_{i}^{2}+b\left(\sum _{i = 1}^{n} X_{i}\right)^{2}

is an unbiased estimate of $\sigma^{2}$ , find $a$ and $b$ . Hence show that, in this case,

Q = \sum _{i = 1}^{n} \frac{\left(X_{i}-\bar{X}\right)^{2}}{(1-\rho )(n-1)}

Problem 1b.5

(a)(b)(c)

Let $X_{1}, X_{2}, \ldots , X_{n}$ be independently distributed as $N\left(\mu , \sigma^{2}\right)$ . Define

S^{2} = \frac{1}{n-1} \sum _{i = 1}^{n}\left(X_{i}-\bar{X}\right)^{2}

and

Q = \frac{1}{2(n-1)} \sum _{i = 1}^{n-1}\left(X_{i+1}-X_{i}\right)^{2}

?

(a)

Prove that $\operatorname {Var}\left[S^{2}\right] = 2 \sigma^{4} /(n-1)$ .

(b)

Show that $Q$ is an unbiased estimate of $\sigma^{2}$ .

(c)

Find the variance of $Q$ and hence show that as $n \rightarrow \infty$ , the efficiency of $Q$ relative to $S^{2}$ is $\frac{2}{3}$ .