The Four Types of Estimable Functions

Estimability

For linear models such as

$Y = X {\beta}+ {\epsilon}$

with $E(Y)={X \beta}$ , a primary analytical goal is to estimate or test for the significance of certain linear combinations of the elements of ${{\beta}}$ .This is accomplished by computing linear combinations of the observed Ys. An unbiased linear estimate of a specific linear function of the individual $\beta$ s, say $L {\beta}$ , is a linear combination of the Ys that has an expected value of $L {\beta}$ . Hence, the following definition:

A linear combination of the parameters $L {\beta}$ is estimable if and only if a linear combination of the Ys exists that has expected value $L {\beta}$ .

Any linear combination of the Ys, for instance KY, will have expectation $E({KY})={KX} {\beta}$ .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 ${{\beta}}$ . Therefore,

$L {\beta}$ 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 $L {\beta}$ is estimable.

Once an estimable L has been formed, $L {\beta}$ 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 $L {\beta}$ in the sense of having minimum variance as well as maximum likelihood when the residuals are normal. To test the hypothesis that $L {\beta}=0$ , compute SS $(H_0\colon L {\beta}=0)=({Lb})^'(L ({X^'X})^{-}L^')^{-1}{Lb}$ and form an F test using the appropriate error term.

Estimability

General Form of an Estimable Function

Introduction to Reduction Notation

Examples

Using Symbolic Notation