##### STAT 5102 HW4 Problem 2

Question Problem 2. Suppose x1, . . . , xn are samples from a distribution with mean

Âµ and variance ?

2

. We consider a class of estimators of Âµ using the weighted

average of the samples

ÂµË†(c1, c2, . . . , cn) = Xn

i=1

ciXi

.

where ci are constants and sum to 1

Xn

i=1

ci = 1.

The sample mean XÂ¯ is Ë†Âµ(

1

n

, . . . ,

1

n

).

(a). Show that Ë†Âµ(c1, c2, . . . , cn) is an unbiased estimator of Âµ.

(b). Compute the following ratio

R =

MSE(Ë†Âµ(c1, c2, . . . , cn))

MSE(XÂ¯)

.

(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

.

Âµ and variance ?

2

. We consider a class of estimators of Âµ using the weighted

average of the samples

ÂµË†(c1, c2, . . . , cn) = Xn

i=1

ciXi

.

where ci are constants and sum to 1

Xn

i=1

ci = 1.

The sample mean XÂ¯ is Ë†Âµ(

1

n

, . . . ,

1

n

).

(a). Show that Ë†Âµ(c1, c2, . . . , cn) is an unbiased estimator of Âµ.

(b). Compute the following ratio

R =

MSE(Ë†Âµ(c1, c2, . . . , cn))

MSE(XÂ¯)

.

(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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