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Model checks for generalized linear models. (English) Zbl 1035.62073
For a generalized regression model \(E(Y\,|\, X=x)=m (\beta^T_0 x, \vartheta_0)\) (\(m\) being a link function, \(\beta_0\), \(\vartheta_0\) are unknown parameters) a test is proposed for the correctness of link function specification. It is based on the residual cusum process projected onto the \(\beta_n\) direction (\(\beta_n\) is some estimator for \(\beta_0\)). Convergence of this process to a Gaussian limit is demonstrated. An innovation process transform is proposed to derive a statistic which is asymptotically distribution free. Results of simulations are presented.

MSC:
62J12 Generalized linear models (logistic models)
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
Software:
Fahrmeir
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[1] Aranda-Ordaz F. S., Biometrika 68 pp 357– (1981)
[2] Cheng K. F., J. Amer. Statist. Assoc 89 pp 657– (1994)
[3] 3L. Fahrmeir, and G. Tutz (1994 ).Multivariate statistical modelling based on generalized linear models. Springer Verlag, New York. · Zbl 0809.62064
[4] 4K.T. Fang, S. Kotz, and K. W. Ng (1990 ).Symmetric multivariate and related distributions. Chapman & Hall, London. · Zbl 0699.62048
[5] Fienberg S. E., J. Amer. Statist. Assoc 79 pp 72– (1984) · doi:10.1080/01621459.1984.10477064
[6] Hardle W., Ann. Statist 21 pp 1926– (1993)
[7] Hardle W., J. Amer. Statist. Assoc 84 pp 986– (1989)
[8] Khmaladze E. V., Theory Probab. Appl 26 pp 240– (1981)
[9] Landwehr J. M., J. Amer. Statist. Assoc 79 pp 61– (1984)
[10] 10P. McCullagh, and J. A. Nelder (1989 ).Generalized linear models, 2nd edn. Chapman & Hall, London. · Zbl 0744.62098
[11] DOI: 10.1016/S0167-7152(96)00081-8 · Zbl 1003.62540 · doi:10.1016/S0167-7152(96)00081-8
[12] Pregibon D., Appl. Statist. 29 pp 15– (1980)
[13] Prentice R. L., Biometrics 32 pp 761– (1976)
[14] DOI: 10.1214/aos/1031833666 · Zbl 0926.62035 · doi:10.1214/aos/1031833666
[15] Stute W., J. Amer. Statist. Assoc. 93 pp 141– (1998)
[16] DOI: 10.1214/aos/1024691363 · Zbl 0930.62044 · doi:10.1214/aos/1024691363
[17] Su J. Q., J. Amer. Statist. Assoc. 86 pp 420– (1991)
[18] Tsiatis A. A., Biometrika 67 pp 250– (1980)
[19] 19A. W. Van Der Vaart, and J. A. Wellner (1996 ).Weak convergence and empirical processes, with applications to statistics.Springer Verlag, New York. · Zbl 0862.60002
[20] Wu C. F. J., Ann. Statist. 14 pp 1261– (1986)
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