Let X be a normally distributed variable with mean and vari

Let X be a normally distributed variable with mean ? and variance ?. A random sample of three observations was obtained from this population. Consider the following estimators of ?.

a. Derive the expected value and variance of each estimator.

b. Compare the properties of these estimators in terms of unbiasedness, efficiency, and

consistency.

Solution

Given that U₁ =

                     U₂     =    +     +   

Apply expectations on both sides we have

E(U₁ ) = E( )

                = E(x₁+x₂₊x₃)

                = (

                =

                =

E(U₂) = E( +     +   )

                = E( )+ E( ) + E()

                = E( x₁) +E(x₂) + E(x₃)

                =  + +

                =

Now apply variance on both sides

var(U₁ ) = var ( )

                = var(x₁+x₂₊x₃)

                = ()

               = 3

                =   

Var (U₂) = var ( +     +   )

                = var ( )+ var ( ) + var ()

                = var( x₁) +var (x₂) + var(x₃)

                =  + +

=

Both the properties are unbiased i.e., the expected value of the statistic is equal to the parameter value

Bothe the properties are efficiency i.e., U1 has the minimum variance than U2

Therefore var(U1) var(U2) and consistency under probability condition