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.


62J12 Generalized linear models (logistic models)
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference


Full Text: DOI


[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)
[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
[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
[15] Stute W., J. Amer. Statist. Assoc. 93 pp 141– (1998)
[16] DOI: 10.1214/aos/1024691363 · Zbl 0930.62044
[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.