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, ∞). 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 θ is

f(x, y) = f(θ) + f′_{x}(θ)(x − θ_{x}) + f′_{y}(θ)(y − θ_{y}) + remainder

Let θ = (EX, EY). The simplest approximation for f(X, Y) is then f(X, Y) = f(θ) + f′_{x}(θ)(X − E(X)) + f′_{y}(θ)(Y − E(Y)) + O(n^{−1})

The approximation for E(f(X, Y)) is therefore

E(f(X, Y)) = E [f(θ) + f′_{x}(θ)(X − E(X)) + f′_{y}(θ)(Y − E(Y)) + O(n^{−1})]
≈ E [f(θ)] + E [f′_{x}(θ)(X − E(X))] + E [f′_{y}(θ)(Y − E(Y))]
= E [f(θ)] + f′_{x}(θ)E [(X − E(X))] + f′_{y}(θ)E [(Y − E(Y))]
= E [f(θ)] + 0 + 0
≈ f(EX, EY)
= EX/EY

The second order Taylor expansion is

f(x, y) = f(θ) + f′_{x}(θ)(x − θ_{x}) + f′_{y}(θ)(x − θ_{y})
+ \frac{1}{2} \{f′′_{xx}(θ)(x − θ_{x})^{2} + 2f′′_{xy}(θ)(x − θ_{x})(y − θ_{y}) + f′′_{yy}(y − θ_{y})^{2}\} + remainder

So a better approximation is for E[f(X, Y)] expanded around θ = (E(X), E(Y)) is

E(f(X, Y)) = f(θ) + \frac{1}{2} \{f′′_{xx}(θ)Var(X) + 2f′′_{xy}(θ)Cov(X, Y) + f′′_{yy}(θ)Var(Y)\} + O(n^{−1})

For f = R/S, f′′_{RR} = 0, f′′_{RS} = −S^{−2}, f′′_{SS} = \frac{2R}{S^{3}}. Then E(R/S) is approximately

E(R/S) ≡ E(f(R, S)) ≈ \frac{ER}{ES} − \frac{Cov(R, S)}{E^{2}S} + \frac{Var(S)ER}{E^{3}S}

By the definition of variance, the variance of f(X, Y) is

Var(f(X, Y)) = E {[f(X, Y) − E(f(X, Y))]^{2}}

Using E(f(X, Y)) ≈ f(θ) (from above)

Var(f(X, Y)) ≈ E {[f(X, Y) − f(θ)]^{2}}

Then using the first order Taylor expansion for f(X, Y) expanded around θ

Var(f(X, Y)) ≈ E {[f(θ) + f′_{x}(θ)(X − θ_{x}) + f′_{y}(θ)(Y − θ_{y}) − f(θ)]^{2}}
= E {f_{x}^{′2}(θ)(X − θ_{x})^{2} + f_{y}^{′2}(θ)(Y − θ_{y})^{2} + 2f′_{x}(θ)(X − θ_{x})f′_{y}(θ)(Y − θ_{y})}
= f_{x}^{′2}(θ)Var(X) + 2f′_{x}(θ)f′_{y}(θ)Cov(X, Y) + f_{y}^{′2}(θ)Var(Y)

Now we return to our example: f(R, S) = R/S. Since f′_{R} = S^{−1}, f′_{S} = \frac{−R}{S^{2}} and θ = (ER, ES), f′_{R}(θ)f′_{S}(θ) = \frac{−ER}{E^{3}S}, and f_{R}^{′2}(θ) = \frac{1}{E^{2}S} , f_{S}^{′2}(θ) = \frac{E^{2}R}{E^{4}S}. and so

Var(R/S) ≈ \frac{1}{E^{2}S} Var(R) + 2\frac{−ER}{E^{3}S}Cov(R, S) + \frac{E^{2}R}{E^{4}S}VarS
= \frac{E^{2}R}{E^{2}S} [\frac{Var(R)}{E^{2}R} − 2\frac{Cov(R,S)}{ER ES} + \frac{Var(S)}{E^{2}S}]
= \frac{μ_{R}^{2}}{μ_{S}^{2}} [\frac{σ_{R}^{2}}{μ_{R}^{2}} − 2\frac{Cov(R,S)}{μ_{R} μ_{S}} + \frac{σ_{S}^{2}}{μ_{S}^{2}} ]

Reference: Kendall’s Advanced Theory of Statistics, Arnold, London, 1998, 6th Edition, Volume 1, by Stuart & Ord, p. 351. Reference: Survival Models and Data Analysis, John Wiley & Sons NY, 1980, by Elandt-Johnson and Johnson, p. 69.