Example 5.43
Here h(µX , µY ) =
µX
µY
with
d
dµX
h(µX , µY ) =
1
µY
and
d
dµY
h(µx, µY ) = −µX
µ2Y
Recall the fist order Taylor approximation:
(1)
Eθh(T ) ≈ h(θ) +
∑
h′i(θ)Eθ(Ti − θi) ≈ h(θ)
since Ti has mean θi.
(2)
Varθh(T ) ≈ Eθ
([
h(T )− h(θ)]2) using above approximation
≈ Eθ
(( k∑
i=1
h′i(θ)(Ti − θi)
)2)
taylor series expansion
=
k∑
i=1
[h′i(θ)]
2Var(Ti) + 2
∑
i>j
h′i(θ)h
′
j(θ)Cov(Ti, Tj)
Approximation (2) is very useful since it provide us with a variance formula for a general
function. By (1) and (2) we get
E
(X
Y
)
=
µX
µY
and
Var
(X
Y
)
≈
( 1
µY
)2
Var(X) +
(
− µX
µ2Y
)2
Var(Y ) + 2
( 1
µY
)(
− µX
µ2Y
)
Cov(X, Y )
=
(µX
µY
)2(Var(X)
µ2X
+
Var(X)
µ2Y
− 2Cov(X, Y )
µXµY
)
We now have an approximation for the mean and variance of the ratio estimator and this
approximation involved the means, variances and covariance of X and Y .
Note: Exact calculation would be quite hopeless with closed form expression being
unattainable.
Now we will find the asymptotic distribution. The inverse of the information matrix is:
I−1n (µX , µY ) =
1
n
(
σ11 σ12
σ21 σ22
)
Applying the delta method gives:
√
n
(X¯
Y¯
− µX
µY
)
d−→ N
(
0,
( 1
µY
,−µX
µ2Y
)(σ11 σ12
σ21 σ22
)( 1
µY−µX
µ2Y
))
2
Now just focusing on the asymptotic variance:
( 1
µY
σ11 − µX
µY
σ21,
σ12
µY
− µX
µY
σ22
)( 1
µY−µX
µ2Y
)
=
σ11
µ2Y
− µX
µ3Y
σ21 +
σ12
µ2Y
− µ
2
X
µ4Y
σ22
=
µ2X
µ2Y
[σ11
µ2X
+
σ22
µ2Y
− 2 σ12
µXµY
]
In other words,
X¯
Y¯
≈ N
(
µX
µY
,
1
n
µ2X
µ2Y
[
σ11
µ2X
+
σ22
µ2Y
− 2 σ12
µXµY
])
3