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Inclusion and exclusion of data or parameters in the general linear model. (English) Zbl 1115.62066
Summary: This paper revisits the topic of how linear functions of observations having zero expectation play an important role in our statistical understanding of the effect of addition or deletion of a set of observations in the general linear model. The effect of adding or dropping a group of parameters is also explained well in this manner. Several sets of update equations were derived by previous researchers in various special cases of the general set-up that we consider here. The results derived here bring out the common underlying principles of these update equations and help integrate these ideas. These results also provide further insights into recursive residuals, design of experiments, deletion diagnostics and selection of subset models.

##### MSC:
 62J05 Linear regression; mixed models 62H12 Estimation in multivariate analysis
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##### References:
 [1] Belsley, D.A.; Kuh, E.; Welsch, R.E., Regression diagnostics: identifying influential data and sources of collinearity. wiley series in probability and mathematical statistics, (1980), Wiley New York · Zbl 0479.62056 [2] Bhaumik, D.; Mathew, T., Optimal data augmentation for the estimation of a linear parametric function in linear models, Sankhyā ser. B, 63, 10-26, (2001) · Zbl 1004.62062 [3] Bhimasankaram, P.; Jammalamadaka, S.R., Recursive estimation and testing in general linear models with applications to regression diagnostics, Tamkang J. math., 25, 353-366, (1994) · Zbl 0816.62049 [4] Bhimasankaram, P.; Jammalamadaka, S.R., Updates of statistics in a general linear model: a statistical interpretation and applications, Comm. statist. simulation comput., 23, 789-801, (1994) · Zbl 0825.62062 [5] Bhimasankaram, P.; Sengupta, D.; Ramanathan, S., Recursive inference in a general linear model, Sankhyā ser. A, 57, 227-255, (1995) · Zbl 0857.62067 [6] Brown, R.L.; Durbin, J.; Evans, J.M., Methods of investigating whether a regression relationship is constant over time (with discussion), J. roy. statist. soc. ser. B, 37, 149-192, (1975) · Zbl 0321.62063 [7] Chambers, J.M., Updating methods for linear models for the addition or deletion of observations, (), 53-65 [8] Chib, S.; Jammalamadaka, S.R.; Tiwari, R., Another look at some results on the recursive estimation in the general linear model, Amer. statist., 41, 56-58, (1987) [9] Cook, R.D.; Weisberg, S., An introduction to regression graphics. wiley series in probability and mathematical statistics, (1994), Wiley New York [10] Farebrother, R.W., Linear least squares computations, (1988), Mercel Dekker New York · Zbl 0715.65027 [11] Gragg, W.B.; LeVeque, R.J.; Trangenstein, J.A., Numerically stable methods for updating regressions, J. amer. statist. assoc., 74, 161-168, (1979) · Zbl 0398.62058 [12] Haslett, S., Updating linear models with dependent errors to include additional data and/or parameters, Linear algebra appl., 237/238, 329-349, (1996) · Zbl 0843.62072 [13] Haslett, S., Recursive estimation of the general linear model with dependent errors and multiple additional observations, Austral. J. statist., 27, 183-188, (1985) · Zbl 0568.62064 [14] Haslett, J., A simple derivation of deletion diagnostic results for the general linear model with correlated errors, J. roy. statist. soc. ser. B, 61, 603-609, (1999) · Zbl 0924.62076 [15] Jammalamadaka, S.R.; Sengupta, D., Changes in the general linear model: a unified approach, Linear algebra appl., 289, 225-242, (1999) · Zbl 0933.62060 [16] Kala, R.; Klaczyński, K., Recursive improvement of estimates in a gauss – markov model with linear restrictions, Canad. J. statist., 16, 301-305, (1988) · Zbl 0667.62050 [17] Kianifard, F.; Swallow, W., A review of the development and application of recursive residuals in linear models, J. amer. statist. assoc., 91, 391-400, (1996) · Zbl 0873.62070 [18] Kourouklis, S.; Paige, C.C., A constrained least squares approach to the general gauss – markov linear model, J. amer. statist. assoc., 76, 620-625, (1981) · Zbl 0475.62052 [19] McGilchrist, C.A.; Sandland, R.L., Recursive estimation of the general linear model with dependent errors, J. roy. statist. soc. ser. B, 41, 65-68, (1979) · Zbl 0398.62059 [20] Mitra, S.K.; Bhimasankaram, P., Generalized inverses of partitioned matrices and recalculation of least squares estimators for data and model changes, Sankhyā ser. A, 33, 395-410, (1971) · Zbl 0236.62049 [21] Nieto, F.H.; Guerrero, V.M., Kalman filter for singular and conditional state-space models when the system state and the observational error are correlated, Statist. probab. lett., 22, 303-310, (1995) · Zbl 0813.62085 [22] Placket, R.L., Some theorems in least squares, Biometrika, 37, 149-157, (1950) · Zbl 0041.46803 [23] Pordzik, P.R., A lemma on $$g$$-inverse of the bordered matrix and its application to recursive estimation in the restricted model, Comput. statist., 7, 31-37, (1992) · Zbl 0800.62385 [24] Pordzik, P.R., Adjusting of estimates in general linear model with respect to linear restrictions, Statist. probab. lett., 15, 125-130, (1992) · Zbl 0752.62046 [25] Schall, R.; Dunne, T.T., A unified approach to outliers in the general linear model, Sankhyā ser. B, 50, 157-167, (1988) · Zbl 0656.62078 [26] Sengupta, D., Optimal choice of a new observation in a linear model, Sankhyā ser. A, 57, 137-153, (1995) · Zbl 0857.62075 [27] Sengupta, D.; Jammalamadaka, S.R., Linear models an integrated approach, (2003), World Scientific New Jersey · Zbl 1049.62080 [28] Storer, B.E.; Crowley, J., A diagnostic for Cox regression for and general conditional likelihoods, J. amer. statist. assoc., 389, 139-147, (1985)
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