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Estimating the error distribution in a single-index model. (English) Zbl 1383.62086
Ferger, Dietmar (ed.) et al., From statistics to mathematical finance. Festschrift in honour of Winfried Stute. Cham: Springer (ISBN 978-3-319-50985-3/hbk; 978-3-319-50986-0/ebook). 209-233 (2017).
Summary: This paper addresses the problem of estimating the error distribution in single-index regression models. We estimate the error distribution function with a weighted nonparametric residual empirical distribution function. Our main result is a first order uniform stochastic expansion of the estimator. This expansion makes it possible to derive asymptotically distribution free goodness-of-fit tests about the error distribution. Our approach is to regard the single-index model as a nonparametric regression model, but with estimated covariates (the estimated indices). However, the usual assumption in classical nonparametric regression, that the covariate distribution is quasi-uniform (bounded and bounded away from zero on its compact support), is not reasonable here. We handle this by introducing weights which restrict the estimation of the link function to intervals.
For the entire collection see [Zbl 1383.62010].

##### MSC:
 62G05 Nonparametric estimation 62J12 Generalized linear models (logistic models) 62G20 Asymptotic properties of nonparametric inference
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