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

\[\begin{eqnarray} \text{Var}\left(\frac{\sum_{i=1}^BX_i}{B}\right) &=&\frac{1}{B^2}\sum_{i=1}^B\text{Var}(X_i) + \frac{1}{B^2}\sum_{i\neq j}^B\text{Cov}(X_i, X_j)\non\\ &=&\frac{\sigma^2}{B} + \frac{B-1}{B}\sigma^2\rho\non\\ &=&\sigma^2\rho + \frac{1-\rho}{B}\sigma^2.\non \end{eqnarray}\]

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.