05 04

Stat 5102 homework solution

STAT 5102 HW4 Problem 2
Question Problem 2. Suppose x1, . . . , xn are samples from a distribution with mean
µ and variance ?
2
. We consider a class of estimators of µ using the weighted
average of the samples
µˆ(c1, c2, . . . , cn) = Xn
i=1
ciXi
.
where ci are constants and sum to 1
Xn
i=1
ci = 1.
The sample mean X¯ is ˆµ(
1
n
, . . . ,
1
n
).
(a). Show that ˆµ(c1, c2, . . . , cn) is an unbiased estimator of µ.
(b). Compute the following ratio
R =
MSE(ˆµ(c1, c2, . . . , cn))
MSE(X¯)
.
(c). From (b) we see R only depends on c1, c2, . . . , cn. Consider R as a function
of (c1, c2, . . . , cn) and find the choice of (c1, c2, . . . , cn) that minimizes R.
What is your conclusion?
[hint: you may want to use the following result (see handout 1, page 9)
1
n
Xn
i=1
x
2
i ? (
1
n
Xn
i=1
xi)
2 =
n ? 1
n
S
2 ? 0.
thus for our problem, take ci = xi
1
n
Xn
i=1
c
2
i ? (
1
n
Xn
i=1
ci)
2 =
1
n2
.

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