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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

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