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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.

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