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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 $\theta$ is
$$ f(x,y) = f(\theta) + f'_x(\theta) (x-\theta_x) + f'_y(\theta) (y-\theta_y) + 
  \mbox{remainder}$$

Let $\theta=(EX,EY)$. The simplest approximation for $E(f(X,Y))$ is therefore
$$
  E(f(X,Y))=f(\theta)+f'_x(\theta) (0) + f'_y(\theta) (0) + O(n^{-1}) 
  \approx f(EX,EY)$$

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

So a better approximation is
$$ E(f(X,Y))=f(\theta) + \half \left\{f''_{xx}(\theta) \Var(X)
            +2f''_{xy}(\theta) \Cov(X,Y) + f''_{yy}(\theta) \Var(Y) \right\} 
            + O(n^{-1})$$

For $g=R/S\cs g''_{RR}=0 \cs g''_{RS}=-S^{-2} \cs
  g''_{SS}=\frac{2R}{S^3}$.  Then $E(R/S)$ is approximately
$$ E(R/S) \equiv E(g(R,S))
   \approx \frac{ER}{ES}-\frac{\Cov(R,S)}{E^2S}+\frac{\Var(S)ER}{E^3S}$$

Using the first order Taylor expansion, the variance is 
$$ \begin{array}{rcl}
  \Var(f(X,Y))&=&E\left\{\left[ f(x,y)-E(f(X,Y)) \right]^2\right\} 
        \approx E\left\{\left[ f(x,y)-f(EX,EY) \right]^2\right\}\\
              &=&E\left\{\left[ f'_x(\theta) (X-\theta_x)
                              + f'_y(\theta) (Y-\theta_y) 
            \right]^2\right\} + O(n^{-1}) \\
        &\approx& f'_x(\theta)^2 \Var(X)
                 +2f'_x(\theta) f'_y(\theta) \Cov(X,Y) 
                 + f'_y(\theta)^2 \Var(Y)
   \end{array}$$

Since $g'_R=S^{-1}, g'_S=\frac{-R}{S^2} \cs \Var(R/S)$ is
$$ \begin{array}{rcl}
   \Var(R/S) &\approx& \frac{1}{E^2S}\Var(R)
                +2\frac{-ER}{E^3S}\Cov(R,S)+\frac{E^2R}{E^4S}\Var{S} \\
             &=& \frac{E^2R}{E^2S} \left[
                 \frac{\Var(R)}{E^2R}-2\frac{\Cov(R,S)}{ES\;ER} 
                +\frac{\Var(S)}{E^2S} \right]
    \end{array} $$

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.

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