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Weak convergence of the empirical process of residuals in linear models with many parameters. (English) Zbl 1012.62016

Summary: When fitting, by least squares, a linear model (with an intercept term) with \(p\) parameters to \(n\) data points, the asymptotic behavior of the residual empirical process is shown to be the same as in the single sample problem provided \(p^3 \log^2 (p) /n \to 0\) for any error density having finite variance and a bounded first derivative. No further conditions are imposed on the sequence of design matrices. The result is extended to more general estimates with the property that the average error and average squared error in the fitted values are of the same order as for least squares.

MSC:

62E20 Asymptotic distribution theory in statistics
62J05 Linear regression; mixed models
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[1] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[2] Koul, H. L. (1969). Asymptotic behaviour of Wilcoxon type confidence regions in multiple linear regression. Ann. Math. Statist. 40 1950-1979. · Zbl 0199.53503
[3] Koul, H. L. (1984). Tests of goodness-of-fit in linear regression. Colloq. Math. Soc. János Bolyai 45 279-315. · Zbl 0616.62060
[4] Loynes, R. M. (1980). The empirical distribution function of residuals from generalized regression. Ann. Statist. 8 285-298. · Zbl 0451.62040
[5] Mammen, E. (1996). Empirical process of residuals for high dimension linear models. Ann. Statist. 24 307-335. · Zbl 0853.62042
[6] Meester, S. G. (1984). Testing for normally distributed errors in block design experiments. M.Sc. thesis, Dept. Mathematics and Statistics, Simon Fraser Univ.
[7] Meester, S. G. and Lockhart, R. A. (1988). Testing for normal errors in designs with many blocks. Biometrika 75 569-575. JSTOR: · Zbl 0654.62064
[8] Mukantseva, L. A. (1977). Testing normality in one-dimensional and multi-dimensional linear regression. Theory Probab. Appl. 22 591-602. · Zbl 0388.62058
[9] Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045
[10] Portnoy, S. (1984). Asymptotic behavior of M-estimators of p regression parameters when p2/n is large I. Consistency. Ann. Statist. 12 1298-1309. · Zbl 0584.62050
[11] Portnoy, S. (1985). Asymptotic behavior of M-estimators of p regression parameters when p2/n is large II. Normal approximation. Ann. Statist. 13 1403-1417. · Zbl 0601.62026
[12] Portnoy, S. (1986). Asymptotic behavior of the empiric distribution of M-estimated residuals from a regression model with many parameters. Ann. Statist. 14 1152-1170. · Zbl 0612.62072
[13] Rao, C. R. (1973). Linear Statistical Inference and Its Applications, 2nd ed. Wiley, New York. · Zbl 0256.62002
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