In: Statistics and Probability
(estimate) is an unbiased estimator of .
Proof: = E(^)= E(Ci yi)
= Ci E(yi)
= Ci( o + 1Xi)
= oCi + 1CiXi equation 1
Now Ci= = = 1
= 0
Substituting values of and in equation 1
E(o^)= o
Hence o^ is an unbiased estimator ofo.
Var(o^)= Var()
= = =
= )
=
Note that
Var(o^) =
Hence the result.