Ex. 15.1

Ex. 15.1

Derive the variance formula (15.1). This appears to fail if $\rho$ is negative; diagnose the problem in this case.

Soln. 15.1

We have

$$\begin{array}{rcl}\text{Var}\left(\frac{\sum _{i=1}^B{X}_i}{B}\right)& =& \frac{1}{B^2}\sum _{i=1}^B\text{Var}(X_i)+\frac{1}{B^2}\sum _{i\ne j}^B\text{Cov}(X_i,X_j)\\ & =& \frac{{\sigma }^2}{B}+\frac{B-1}{B}{\sigma }^2\rho \\ & =& {\sigma }^2\rho +\frac{1-\rho }{B}{\sigma }^2.\end{array}$$

The assumption implicitly assumes that $\rho \ge -\frac{1}{B-1}$ by noting the variance above is non-negative. When $B$ is large, this (negative) lower bound is close to zero.