Theorem: By respecting assumptions described last time, the estimators and , in the class of unbiased linear estimators, are best linear unbiased estimators of and respectively. That is, they are linear unbiased estimators that have minimium variance.
Last time I showed that is linear. Today, I show that it is unbiased–that the expected value of is equal to .
Proof: Recall that , where , where . Since we assume the values of are fixed, we treat as a constant. Additionally, recall that . Therefore we may write,
Notice now that , since we know the value of from the given sample. Moreover, since the sum of the deviations from the mean is zero, . Thus, . But, we can do better: . Notice that since , . Thus, .
So, what is ? Well, . Notice that , since we are summing over . Thus, . Therefore, . Finally then, .
As a result, . So, taking expectation on both sides, and noting that can be treated as constants since they are non-stochastic, , since by the third assumption (which can be found in part 1).
There we go! Next time I will finish up the proof and show that has minimum variance.