Abstract
As regards the estimation of estimable functions of β and the prediction of predictable functions of β and u, we have, to this point in our development, considered only point estimation and prediction. Now that we have added distribution theory related to the multivariate normal distribution to our arsenal, we may consider interval and “regional” estimation/prediction and hypothesis testing for such functions under normal linear models. Furthermore, we may use hypothesis testing or other procedures that require a normality assumption to choose the linear model upon which further inferences may be based. This chapter takes up all of these topics.
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Notes
- 1.
Moving the derivative inside the integral can be justified by the dominated convergence theorem.
- 2.
More precisely, random variables on a common probability space.
- 3.
More precisely, measurable subsets of a probability space.
References
Burnham, K. P., & Anderson, D. R. (2002). Model selection and multimodel inference: A practical information-theoretic approach (2nd ed.). New York: Springer.
Carlstein, E. (1986). Simultaneous confidence regions for predictions. The American Statistician, 40, 277–279.
Ghosh, B. K. (1973). Some monotonicity theorems for χ 2, F and t distributions with applications. Journal of the Royal Statistical Society, Series B, 35, 480–492.
Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.
Lane, T. P., & Dumouchel, W. H. (1994). Simultaneous confidence intervals in multiple regression. The American Statistician, 48, 315–321.
Lehmann, E., & Romano, J. P. (2005). Testing statistical hypotheses (3rd ed.). New York: Springer.
Scheffé, H. (1959). The analysis of variance. New York: Wiley.
Sidák, Z. (1968). On multivariate normal probabilities of rectangles: Their dependence on correlations. Annals of Mathematical Statistics, 39, 1425–1434.
Tukey, J. W. (1949a). Comparing individual means in the analysis of variance. Biometrics, 5, 99–114.
Tukey, J. W. (1949b). One degree of freedom for non-additivity. Biometrics, 5, 232–242.
Zimmerman, D. L. (1987). Simultaneous confidence regions for predictions based on the multivariate t distribution. The American Statistician, 41, 247.
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Zimmerman, D.L. (2020). Inference for Estimable and Predictable Functions. In: Linear Model Theory. Springer, Cham. https://doi.org/10.1007/978-3-030-52063-2_15
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DOI: https://doi.org/10.1007/978-3-030-52063-2_15
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