Abstract
Now that we have determined which functions c Tβ in a linear model {y, Xβ} can be estimated unbiasedly, we can consider how we might actually estimate them. This chapter presents the estimation method known as least squares. Least squares estimation involves finding a value of β that minimizes the distance between y and Xβ, as measured by the squared length of the vector y −Xβ. Although this seems like it should lead to reasonable estimators of the elements of Xβ, it is not obvious that it will lead to estimators of all estimable functions c Tβ that are optimal in any sense. It is shown in this chapter that the least squares estimator of any estimable function c Tβ associated with the model {y, Xβ} is linear and unbiased under that model, and that it is, in fact, the “best” (minimum variance) linear unbiased estimator of c Tβ under the Gauss–Markov model {y, Xβ, σ 2I}. It is also shown that if the mean structure of the model is reparameterized, the least squares estimators of estimable functions are materially unaffected.
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References
Harville, D. A. (1997). Matrix algebra from a statistician’s perspective. New York: Springer.
Snedecor, G. W., & Cochran, W. G. (1980). Statistical methods (7th ed.). Iowa: Iowa State University Press.
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Zimmerman, D.L. (2020). Least Squares Estimation for the Gauss–Markov Model. In: Linear Model Theory. Springer, Cham. https://doi.org/10.1007/978-3-030-52063-2_7
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DOI: https://doi.org/10.1007/978-3-030-52063-2_7
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