zbMATH — the first resource for mathematics

On the use of nonparametric regression for model checking. (English) Zbl 0663.62096
To asses the fit of a parametric nonlinear regression model confidence bands around the fitted regression curve are compared with those around the curve fitted by nonparametric regression. Further, a pseudo likelihood ratio test is developed. The idea is first demonstrated for binary (logistic) regression with a single covariate and then for Poisson regression. In the last chapter a general approach is discussed.
Let f(.,r(x),\(\psi)\) be the conditional distribution of y for given x, r(x,\(\vartheta)\) a parametric model of r(x), \({\hat \vartheta}^ a \)consistent estimator of \(\vartheta\), \({\hat \psi}\) an estimator of \(\psi\) and \(\hat r(x)\) a nonparametric estimate of r(x) based on n observations each. For a global comparison of the two curves r(x,\({\hat \vartheta}\)) and \(\hat r(x)\) the test statistic \[ \sum^{n}_{i=1}\{\log f(y_ i,\hat r(x_ i),{\hat \psi})-\log f(y_ i,r(x_ i,{\hat \vartheta}),{\hat \psi})\} \] is proposed.
Reviewer: D.Rasch

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62J02 General nonlinear regression
Full Text: DOI