

\subsection{\bf Error in Lack of}
\subsection{\bf Correlation Assumption}

\begin{slide}
\begin{itemize}
\item If $y=a_1 x_1 + a_2 x_2$, 
\item then $\sigma^2_y = a_1^2 \sigma_1^2 + a_2^2 
\sigma_2^2$ $ + 2a_1 a_2 \sigma_1 \sigma_2 \rho_{12}$, 
\item and all one can say in general is that
\[
\sigma^2_y \leq \left( | a_1 \sigma_1 | + | a_2 \sigma_2 | \right)^2
\]
\item The uncorrelated estimate is
\[
\tilde\sigma_y = \sqrt{a_1^2 \sigma_1^2 + a_2^2 \sigma_2^2}
\]
\item and the true $\sigma_y$ satisfies
\[
0\leq \sigma_y \leq \sqrt{2} \tilde\sigma_y
\]
\item where extremes are only attained when $| a_1 \sigma_1 | = |a_2
\sigma_2|$ which is a most dangerous case.

\item Other ratios, $|a_1 \sigma_1|/|a_2 \sigma_2| \neq 1$ are better,
e.g., $a_1 \sigma_1 = 0 \Rightarrow \sigma_y = \tilde\sigma_y$!

\item For $n$ variables, all one can say rigorously is
\[
0\leq \sigma_y \leq \sqrt{n} \tilde\sigma_y
\]
which is clearly pretty bad for large $n$.
\end{itemize}
\end{slide}

