1 Suppose we have two unbiased estimators T and of E Ri a Sh

1. Suppose we have two unbiased estimators T and of E Ri. (a) Show that the combined estimator T a Ti (1-a)T is also an unbiased estimator whenever a E 0, 1 (b) If Ti and T2 are determined from independent samples, then calculate Var(T) in terms of v1 Var (T1) and v2 Var (T2) (c For the situation given in (b), determine the best choice of a in the sense that for such a choice Var(T) is smallest. What is the effect on this combined estimator of Ti having a very large variance relative to T2?

Solution

a)

As T₁ and T₂ is unbiased estimator of , thus

E(T₁) = and E(T₂) =

Now we are given that

T = T_1*+ (1-)T₂

T can be said as unbiased estiomator if E(T) =  

E(T) = E(T₁)_*+ (1-) E(T₂)

= * +(1-)*

=( - ) +

=

As E(T) =  , therefore T is also a unbiased estimator of  

b)

Given that T = T_1*+ (1-)T₂

Var(T) = ² Var(T₁) + (1-)²* Var(T₂) (Covariance term is zero because both T₁ andT₂ are independent of each othre)

Var(T) =  ²v₁+(1+² -2) *v₂

   = (v₁+v₂)² -2 v₂ +v₂

c)

we will choose such that var(T) become smallest by diffrentiating var(T) with respect to

First derrivative of (var(T)) = 2(v₁+v₂) -2v₂

If we put first derrivative of (var(T)) =0

then 2(v₁+v₂) -2v₂ = 0

and = [v₂ / (v₁+v₂) ]

thus = [v₂ / (v₁+v₂) ] is the value that makes var(T) smallest.

Effect of large variance of T₁ relative to variance of T₂ on Var(T)

suppose v₁/v₂ = e>1 (given)

thus v₁= v₂*e

var(T)= (v₁+v₂)² -2 v₂ +v₂

   =(e*v₂+v₂)² -2 v₂ +v₂ (Put v₁= v₂*e)

= v2(e*² + ² -2) +v₂

=(e*[v₂ / (v₁+v₂)]² + [v₂ / (v₁+v₂)]² -2 *[v₂ / (v₁+v₂)] +v₂(put value of = [v₂ / (v₁+v₂) ] )

= (e*v₂^{2 +}v₂² -2 *v₂ *(v₁+v₂) / (v₁+v₂)² + v₂

=(e*v₂^{2 +}v₂²  -2 *v₂²_*e - 2 *v₂²) / (v₁+v₂)² + v₂(again put v₁= v₂*e in numerator)

= - (e*v₂² +v₂²) / (v₁+v₂)² +v₂

or

var(T) =  - (e*v₂² +v₂²) / (v₁+v₂)² +var (T₂)

therfore it can be said that

var(T) < Var(T₂) ( equation 1)

we are already given that var(T₁) > var(T₂)    ( equation 2)

from equation 1 and 2

it can be concluded that

var(T) < Var(T2) < var(T1)

since variance of T is smallest in all three estimators, thus it can be

interpreted that T is best esimator in all three if var(T₁) >>var(T₂)