Estimability

Abstract

In this chapter, it is not yet our objective to learn how to estimate the parameters of a linear model (or functions thereof) that are of interest, but rather to learn whether it is even possible to “sensibly estimate” those parameters or functions. Precisely what is meant by “sensibly estimate”? Several definitions are possible, but in this book the phrase is synonymous with “estimate unbiasedly.” Parameters, or functions thereof, that can be estimated unbiasedly from the available data (X and y) are said to be estimable, with the remainder designated as nonestimable. In this chapter we focus exclusively on the estimability of linear functions of β, i.e., functions of the form c ^Tβ where c is a specified, nonrandom p-vector. Later it will be shown that the estimability of a linear function is necessary and sufficient for a “best” (minimum variance) linear unbiased estimator of that function to exist, and we will eventually derive that estimator. We defer the issue of estimability for variance–covariance parameters (or functions thereof) to Chap. 16.