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) + fx′ (θ)(x − θx ) + fy′ (θ)(y − θy ) + remainder
Let θ = (EX, EY ). The simplest approximation for f (X, Y ) is then
f (X, Y ) = f (θ) + fx′ (θ)(X − E(X)) + fy′ (θ)(Y − E(Y )) + O(n−1)
The approximation for E(f (X, Y )) is therefore
E(f (X, Y )) =
≈
=
=
≈
=
h
E f (θ) + fx′ (θ)(X − E(X)) + fy′ (θ)(Y − E(Y )) + O(n−1 )
h
i
E [f (θ)] + E [fx′ (θ)(X − E(X))] + E fy′ (θ)(Y − E(Y ))
E [f (θ)] + fx′ (θ)E [(X − E(X))] + fy′ (θ)E [(Y − E(Y ))]
E [f (θ)] + 0 + 0
f (EX, EY )
EX/EY
i
The second order Taylor expansion is
′
′
f (x, y) = f (θ)
o
n + fx (θ)(x − θx ) + fy (θ)(x − θy )
′′
′′
′′
+ 21 fxx
(θ)(x − θx )2 + 2fxy
(θ)(x − θx )(y − θy ) + fyy
(y − θy )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) +
o
1 n ′′
′′
′′
fxx (θ)Var(X) + 2fxy
(θ)Cov(X, Y ) + fyy
(θ)Var(Y ) + O(n−1 )
2
′′
′′
′′
For f = R/S, fRR
= 0, fRS
= −S −2 , fSS
=
E(R/S) ≡ E(f (R, S)) ≈
2R
.
S3
Then E(R/S) is approximately
ER Cov(R, S) Var(S)ER
−
+
ES
E2S
E3S
By the definition of variance, the variance of f (X, Y ) is
n
Var(f (X, Y )) = E [f (X, Y ) − E(f (X, Y ))]2
Using E(f (X, Y )) ≈ f (θ) (from above)
n
Var(f (X, Y )) ≈ E [f (X, Y ) − f (θ)]2
o
o
Then using the first order Taylor expansion for f (X, Y ) expanded around θ
Var(f (X, Y )) ≈ E
h
f (θ) + fx′ (θ)(X − θx ) + fy′ (θ)(Y − θy ) − f (θ)
n
i2 = E fx′2 (θ)(X − θx )2 + fy′2 (θ)(Y − θy )2 + 2fx′ (θ)(X − θx )fy′ (θ)(Y − θy )
= fx′2 (θ)Var(X) + 2fx′ (θ)fy′ (θ)Cov(X, Y ) + fy′2 (θ)Var(Y )
o
Now we return to our example: f (R, S) = R/S.
Since fR′ = S −1 , fS′ =
E2R
.
E4S
−R
S2
and θ = (ER, ES), fR′ (θ)fS′ (θ) =
−ER
,
E3S
and fR′2 (θ) =
1
,
E2S
fS′2 (θ) =
and so
Var(R/S) ≈
=
=
2
1
Var(R) + 2 −ER
Cov(R, S) + EE4R
VarS
E2S E3S
S
Cov
(R,S)
Var
(S)
E 2 R Var(R)
− 2 ER ES + E 2 S
E2S
E2R
µ2R
µ2S
2
σR
2
µR
(R,S)
− 2 Cov
+
µR µS
2
σS
2
µS
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 ElandtJohnson and Johnson, p. 69.