MS&E 223
Lecture Notes #8
Simulation
Estimating Nonlinear Functions of Means
Brad Null
Spring Quarter 200809
Page 1 of 7
Estimating Nonlinear Functions of Means
Back at the beginning of the course, we discussed how to obtain point estimates and confidence intervals
for quantities of the form
μ
= E[X]. (E.g., X is the random reward from a play of the gambling game.) In
many applications, we need to estimate quantities of the form
1
2
d
g(
,
,
,
)
α =
μ μ
μ
K
, where g is a nonlinear
function and
(i)
i
E[X ]
μ =
for 1
≤
i
≤
d.
To keep notation simple, we will focus on the case k = 2, and look at quantities of the form
X
Y
g(
,
)
α =
μ
μ
with
X
E[X]
μ =
and
Y
E[Y]
μ =
.
1.
Examples
Example 1
:
Suppose that one is running a retail outlet in which one wishes to compute the longrun
average revenue generated per customer.
Let
i
R denote the revenue generated on day i, and set
i
i
X
R
=
i
Y
=
total number of customers on day i.
It is often reasonable to assume that the
i
i
(X ,Y ) pairs are i.i.d. as a random pair (X,Y). Then, the average
revenue over n days is equal to
n
n
n
1
n
1
Y
X
=
Y
+
...
Y
X
+
...
X
+
+
where
∑
=
=
n
1
i
i
n
X
n
1
X
and
∑
=
=
n
1
i
i
n
Y
n
1
Y
.
Observe that
Y
X
n
n
Y
X
μ
μ
→
with probability 1 as n
→
∞
by the SLLN (applied to both the numerator and demoninator).
Thus, the longrun average revenue per customer,
α,
takes the form
X
Y
g(
,
)
α =
μ
μ
, where g(x, y) = x/y.
Here, the (X
i
, Y
i
) replicates can be simulated by repeatedly and independently simulating one day’s
operation of the retail outlet.
Example 2
:
Suppose that one wishes to estimate the variance of the revenue generated per day. As
above, suppose that
{ }
i
R :i 1
≥
are i.i.d. as a random variable R. Set
X =
2
R
and
Y = R
Then we wish to compute
( 29
2
2
X
Y
E[R ]
E[R]
g(
,
)
α =

=
μ
μ
, where
2
g(x,y)
x
y
=

.