The Four Types of Estimable Functions

# Estimability

For linear models such as with , a primary analytical goal is to estimate or test for the significance of certain linear combinations of the elements of .This is accomplished by computing linear combinations of the observed Ys. An unbiased linear estimate of a specific linear function of the individual s, say , is a linear combination of the Ys that has an expected value of . Hence, the following definition:
A linear combination of the parameters is estimable if and only if a linear combination of the Ys exists that has expected value .
Any linear combination of the Ys, for instance KY, will have expectation .Thus, the expected value of any linear combination of the Ys is equal to that same linear combination of the rows of X multiplied by . Therefore, is estimable if and only if there is a linear combination of the rows of X that is equal to L -that is, if and only if there is a K such that L = KX.
Thus, the rows of X form a generating set from which any estimable L can be constructed. Since the row space of X is the same as the row space of X'X, the rows of X'X also form a generating set from which all estimable Ls can be constructed. Similarly, the rows of (X'X)-X'X also form a generating set for L.

Therefore, if L can be written as a linear combination of the rows of X, X'X, or (X'X)-X'X, then is estimable.

Once an estimable L has been formed, can be estimated by computing Lb, where b = (X'X)-X'Y. From the general theory of linear models, the unbiased estimator Lb is, in fact, the best linear unbiased estimator of in the sense of having minimum variance as well as maximum likelihood when the residuals are normal. To test the hypothesis that , compute SS and form an F test using the appropriate error term.