# Statistics theory of estimation

Egerton University

Mathematics Department

STAT 332: Theory of Estimation

Problem Set 3

1. Show that if T is an unbiased estimation of

ο± , then aT + b is unbiased estimator of aο± + b . Is T 2

an unbiased estimator

of ο± ?

2. Consider the random variables π1 , π2 , β― , ππ , where π β₯ 10, πΈ(ππ ) = π, πππ(ππ ) = π 2 > 0, and the estimator πΜ π =

1

βππ=1 ππ . Calculate

2

πβ10

a) The bias of πΜ π

b) The variance of πΜ π

c) The MSE of πΜ π

3. Let π1 , π2 , β― , ππ be a random sample from Bernoulli

( p) with common density f (x; p ) = p x (1 β p )1β x , x = 0,1 , where

0 οΌ p οΌ 1 is unknown.

1 n

a) Show that X = ο₯ X i is an unbiased estimator of p

n i =1

p (1 β p )

b) Show that Var ( X ) =

n

ο© X (1 β X ) οΉ

ο© p (1 β p ) οΉ

c) Show that E οͺ

οΊ = (1 β p ) οͺ

οΊ.

n

n

οͺο«

οΊο»

ο«

ο»

d) Find the value of

4. Let X 1 , X 2 ,

c

(

so that cX 1 β X

) is an unbiased estimator of Var ( X ) =

, X n be a random sample from f ( x;ο± ) =

1

ο±

1

p (1 β p )

n

.

β1

x ο± , 0 οΌ x οΌ 1, where ο± οΎ 0 .is unknown.

1 n

a) Show that the MLE of ο± is ο±Λ = β ο₯ ln X i .

n i =1

b) Show that ο±Λ is mean-squared error consistent estimator of ο± .

5. Let X 1 , X 2 , , X n be a random sample from π(π₯; π) = π β(π₯βπ) , π₯ > π, where ο¬ οΎ 0 is unknown.

a) Show that the MME of π is πβ β 1.

b) Is the MME consistent?

6. Let X 1 , X 2 ,

, X n be IID gamma (ο‘ , ο’ ) with density f (x;ο‘ , ο’ ) =

ο’ ο‘ ο‘ β1 β ο’x

x e , x οΎ 0 , where ο‘ is known but ο’

ο(ο‘ )

is

unknown.

a) Find the MLE ο’Λ of

b)

ο’ .

Show that 1 ο’Λ is unbiased estimator of

7. Let X 1 , X 2 ,

1 ο’ ..

, X n be a random sample from f (x; ο¬ ) = ο¬e βο¬x , x οΎ 0 , where ο¬ οΎ 0 is unknown.

E (1 X ) and then construct an unbiased estimator of ο¬ .

b) Find the MSE of 1 X .

a) Compute

8. Let

Tn be a statistic such that lim E (Tn ) = ο± and lim ο³ T2n = 0 . Prove that Tn is a consistent estimator of ο± .

nβο₯

nβο₯

, X n be a random sample from a uniform distribution on the interval (ο± , 2ο± ) ,where ο± οΎ 0 is unknown.

n +1

ο2 X (n ) + X (1) οis an unbiased estimator of ο± .

Shown that T =

5n + 4

9. Let

X1 , X 2 ,

Name:

Description:

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