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\centerline{\bf \large Approximations for Mean and Variance of a Ratio}

Consider random variables $R$ and $S$ where $S$ either has no mass at
0 (discrete) or has support $[0,\infty)$.  Let $G=g(R,S)=R/S$.  Find
approximations for $EG$ and $\Var(G)$ using Taylor expansions of
$g()$.

For any $f(x,y)$, the bivariate first order Taylor expansion about any
$\bm{\theta} = (\theta_x, \theta_y)$ is
\begin{equation}
f(x,y) = f(\bm{\theta}) + f'_x(\bm{\theta}) (x-\theta_x) + 
   f'_y(\bm{\theta}) (y-\theta_y) + \mbox{R}
\end{equation}
where R is a remainder of smaller order than the terms in the equation.

Switching to random variables with finite means $EX \equiv \mu_x$ and
$EY \equiv \mu_y$, we can choose the expansion point to be
$\bf{\theta}=(\mu_x,\mu_y)$. In that case the first order Taylor series
approximation for $f(X,Y)$ is
\begin{equation}
  f(X,Y)=f(\bm{\theta})+f'_x(\bm{\theta}) (X-\mu_x) + f'_y(\bm{\theta}) (Y-\mu_y) + R 
\end{equation}

The approximation for $E(f(X,Y))$ is therefore
\begin{eqnarray}
  E(f(X,Y)) & = & E\left[f(\bm{\theta})+f'_x(\bm{\theta}) (X-\mu_x) + f'_y(\bm{\theta}) (Y-\mu_y) + R\right] \\
       & \approx & E\left[f(\bm{\theta})\right]+E\left[f'_x(\bm{\theta}) (X-\mu_x)\right] + E\left[f'_y(\bm{\theta}) (Y-\mu_y)\right]  \\
       & = & E\left[f(\bm{\theta})\right]+f'_x(\bm{\theta}) E\left[(X-\mu_x)\right] + f'_y(\bm{\theta}) E\left[(Y-\mu_y)\right]  \\
       & = & E\left[f(\bm{\theta})\right] + 0  + 0 \\
  &= & f(\mu_x,\mu_y)
\end{eqnarray}


Note that if $f(X,Y)$ is a linear combination of $X$ and $Y$, this result 
matches the well-known result from mathematical statistics that 
$E(aX+bY) = aEX + bEY = a\mu_x + b\mu_y$,
and in
that case the error of approximation is zero.  But with the Taylor
series expansion, we have extended that result to non-linear functions
of $X$ and $Y$.


For our example where $f(x,y) = x/y$ the approximation is 
$E(X/Y) = E(f(X,Y))=f(\mu_x,\mu_y)=\mu_x/\mu_y$.  

The second order Taylor expansion is
\begin{eqnarray}
  f(x,y) &=& f(\bm{\theta}) + f'_x(\bm{\theta}) (x-\theta_x) + f'_y(\bm{\theta}) (y-\theta_y) \\
  &+& 
   \half \left\{f''_{xx}(\bm{\theta}) (x-\theta_x)^2 
              +2f''_{xy}(\bm{\theta}) (x-\theta_x)(y-\theta_y)
               +f''_{yy}(y-\theta_y)^2 \right\} + R
 \end{eqnarray}

So a better approximation is for $E[f(X,Y)]$ expanded around $\bm{\theta}=(\mu_x,\mu_y)$ is
\begin{equation}
E(f(X,Y)) \approx f(\bm{\theta}) + \half \left\{f''_{xx}(\bm{\theta}) \Var(X)
            +2f''_{xy}(\bm{\theta}) \Cov(X,Y) + f''_{yy}(\bm{\theta}) \Var(Y) \right\}.
\end{equation}
Note that we again use the fact that $E(X-\mu_x)=0$, and we now add in the definitions
for variance and covariance:
$\Var(X) = E[(X-\mu_x)^2]$ and $\Cov(X) = E[(X-\mu_x)(Y-\mu_y)]$.

For $f(R,S)=R/S$, the derivatives are $f''_{RR}(R,S)=0 \cs f''_{RS}(R,S)=-S^{-2} \cs \mbox{and }
  f''_{SS}(R,S)=\frac{2R}{S^3}$.

Specifically, when $\bm{\theta}=(\mu_R,\mu_S)$, we have 
$f(\bm{\theta})=\mu_R/\mu_S \cs f''_{RR}(\bm{\theta})=0 \cs 
 f''_{RS}(\bm{\theta})=-\frac{1}{(\mu_S)^2} \cs \mbox{and }
 f''_{SS}(\bm{\theta})=\frac{2\mu_R}{(\mu_S)^3}$.

Then an improved approximation of $E(R/S)$ is
\begin{equation}
E(R/S) \equiv E(f(R,S))
   \approx \frac{\mu_R}{\mu_S}-\frac{\Cov(R,S)}{(\mu_S)^2}+\frac{\Var(S)\mu_R}{(\mu_S)^3}
\end{equation}

By the definition of variance, the variance of $f(X,Y)$ is
\begin{equation}
\Var(f(X,Y)) = E\left\{\left[ f(X,Y)-E(f(X,Y)) \right]^2\right\}
\end{equation}

Using $E(f(X,Y)) \approx f(\bm{\theta})$ (from above)
\begin{equation}
\Var(f(X,Y))  \approx  E\left\{\left[ f(X,Y)- f(\bm{\theta}) \right]^2\right\}
\end{equation}


Then using the first order Taylor expansion for $f(X,Y)$ expanded around $\bm{\theta}$
\begin{eqnarray}
 \Var(f(X,Y))  &\approx&  E\left\{\left[ f(\bm{\theta}) + f'_x(\bm{\theta})(X-\theta_x)+ f'_y(\bm{\theta})(Y-\theta_y) - f(\bm{\theta})  \right]^2\right\} \\
&=&  E\left\{\left[ f'_x(\bm{\theta})(X-\theta_x)+ f'_y(\bm{\theta})(Y-\theta_y))  \right]^2\right\} \\
&=&  E\left\{f'^2_x(\bm{\theta})(X\!-\!\theta_x)^2 \!+\! 
2f'_x(\bm{\theta})(X\!-\!\theta_x) f'_y(\bm{\theta})(Y\!-\!\theta_y)\!+\! 
f'^2_y(\bm{\theta})(Y\!-\!\theta_y)^2  \right\} \\ 
&=& f'^2_x(\bm{\theta}) \Var(X) + 2 f'_x(\bm{\theta}) f'_y(\bm{\theta}) \Cov(X,Y) + f'^2_y(\bm{\theta}) \Var(Y) 
\end{eqnarray}

Now we return to our example: $f(R,S) = R/S$ expanded around $\bm{\theta}=(\mu_R,\mu_S)$.

Since $f'_R=S^{-1}, f'_S=\frac{-R}{S^2}$ and $\bm{\theta}=(\mu_R, \mu_S)$, we now have
$f'^2_R(\bm{\theta}) = \frac{1}{(\mu_S)^2} \cs 
f'_R(\bm{\theta}) f'_S(\bm{\theta}) = \frac{-\mu_R}{(\mu_S)^3} \cs
f'^2_S(\bm{\theta}) = \frac{(\mu_R)^2}{(\mu_S)^4}$.


and so
\begin{eqnarray}
   \Var(R/S) &\approx& \frac{1}{(\mu_S)^2}\Var(R)
                +2\frac{-\mu_R}{(\mu_S)^3}\Cov(R,S)+\frac{(\mu_R)^2}{(\mu_S)^4}\Var(S) \\
             &=& \frac{(\mu_R)^2}{(\mu_S)^2} \left[
                 \frac{\Var(R)}{(\mu_R)^2}-2\frac{\Cov(R,S)}{\mu_R \; \mu_S} 
                +\frac{\Var(S)}{(\mu_S)^2} \right] \\
             &=& \frac{(\mu_R)^2}{(\mu_S)^2} \left[
                 \frac{\sigma_R^2}{(\mu_R)^2} -2\frac{\Cov(R,S)}{\mu_R\;\mu_S} 
                +\frac{\sigma_S^2}{(\mu_S)^2} \right]
    \end{eqnarray}

Reference: {\it Kendall's Advanced Theory of Statistics}, Arnold, London, 1998,
$6^{\mbox{th}}$ Edition, Volume 1, by Stuart \& Ord, p. 351.

Reference: {\it Survival Models and Data Analysis}, John Wiley \& Sons NY, 1980,
by Elandt-Johnson and Johnson, p. 69.

\end{document}



