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Goodness-of-fit tests in parametric regression based on the estimation of the error distribution. (English) Zbl 1196.62049
Summary: Consider a heteroscedastic regression model $$Y=m(X)+\sigma (X)\varepsilon$$, where $$m(X)=E(Y|X)$$ and $$\sigma ^{2}(X)= \text{Var} (Y|X)$$ are unknown, and the error $$\varepsilon$$ is independent of the covariate $$X$$. We propose a new type of test statistic for testing whether the regression curve $$m(\cdot )$$ belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.

##### MSC:
 62G10 Nonparametric hypothesis testing 62G30 Order statistics; empirical distribution functions 62G20 Asymptotic properties of nonparametric inference 62F05 Asymptotic properties of parametric tests
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