Question

Consider an i.i.d. random sample of size 3 denoted by ?1,?2, ?3 from the same population, where the mean ? and variance ? 2 are unknown. Suppose that you have the following two different estimators for mean ?. (Remember: no work, no credit.) ?̂1 = 0.3?1 + 0.5?2 + 0.2?3 ?̂2 = 0.5?1 + 0.5?3 a. Is ?̂1 unbiased? b. Is ?̂2 unbiased? c. Which one is preferred, ?̂1 or ?̂2?

Answer 1

Solution

Consider an independent and identically distributed random sample X1,X2,X3 of size 3 from the same population with

Mean  E(X)= and Variance V( X) = 2

Consider two different estimators of

1 =0.3 X1 + 0.5 X2 +0.2 X3

E( 1)=E(0.3 X1 + 0.5 X2 +0.2 X3 )

=0.3E(X1) +0.5E(X2)+0.2E(X3)

=0.3 +0.5 +0.2

=

1 is unbiased estimator of .

2=0.5X1 + 0.5 X3

      E( 2)=E(0.5X1 + 0.5 X3)

=0.5E(X1) +0.5E(X3)

=0.5+0.5

=

2is unbiased estimator of .

Now V(1)=V(0.3 X1 + 0.5 X2 +0.2 X3)

=0.322+0.522+0.222

   =0.38 2

Similarly V(2)=V(0.5X1 +0.5X3)

=0.522+0.522

  =0.52

Since V(1) < V(2) then 1  is better estimator. So

1 is preferred nore than 2.