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Diagnostics for assessing regression models. (English) Zbl 0736.62063
Summary: We concern ourselves with diagnostics for checking the overall and local goodness of fit of a model $$s(x)$$ used in the regression of $$Y$$ on $$x\in U=[0,1]^ d$$. The model for $$s(x)$$ is a functional form that depends on a finite number of unknown parameters. Two statistics, $$\Lambda$$ and $$\Lambda_ w$$, are proposed that measure the level of agreement between the model fit to the data and the nonparametric kernel estimator on $$m$$ preselected points in $$U$$. Conditions are given under which $$\Lambda$$ and $$\Lambda_ w$$ are asymptotically equivalent. Both of these statistics measure overall lack of fit and are related to the deviance. Their asymptotic distribution under the null model and under local alternatives is derived. This work is motivated by the local mean deviance plot of J. M. Landwehr, D. Pregibon and A. C. Shoemaker [ibid. 79, 61-83 (1984; Zbl 0531.65080)] for assessing overall lack of fit in logistic regression. Their plot is summarized by our test statistics and is extended to other likelihood based regressions of $$Y$$ on $$x$$.

##### MSC:
 62J20 Diagnostics, and linear inference and regression 62G07 Density estimation
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