Akritas, Michael G.; Van Keilegom, Ingrid Non-parametric estimation of the residual distribution. (English) Zbl 0980.62027 Scand. J. Stat. 28, No. 3, 549-567 (2001). The nonparametric regression model \(Y_i=m(X_i)+\sigma(X_i)\varepsilon_i\) is considered, where \((X_i,Y_i)\) are i.i.d. observed vectors, \(\varepsilon_i\) are independent of \(X_i\), and \(m\) and \(\sigma\) are unknown location and scale functions. The problem is to estimate the distribution function \(F_e\) of \(\varepsilon_i\). The authors suppose that \[ m(x)=\int_0^1 F^{-1}(s |x)J(s)ds,\quad \sigma^2(x)=\int_0^1(F^{-1}(s |x))^2J(s)ds-m^2(x), \] where \(F(s |x)=\Pr\{Y\leq y\;|\;X=x\}\) and \(J\) is a given score function. The function \(F(s |x)\) is estimated by the weighted empirical function \(\tilde F(y |x)=\sum_{i=1}^n W_i(x,a_n){\mathbf 1}\{Y_i\leq y\}\), where \[ W_i(x,a_n)=K\left((x-X_i)/a_n)\right)/ \sum_{j=1}^nK\left((x-X_j)/a_n)\right), \] \(K\) being a kernel and \(a_n\) a bandwidth. Then the estimators \(\hat m\), \(\hat\sigma\) for \(m\) and \(\sigma\) are obtained replacing \(F(y |x)\) by \(\tilde F(y |x)\). The d.f. \(F_e\) is estimated by the empirical d.f. \(\hat F_e\) of the residuals \(\hat \varepsilon_i=(Y_i-\hat m(X_i))/\hat\sigma(X_i) \).The authors derive the convergence rates of \(\hat m\) and \(\hat\sigma\) and demonstrate that \(\sqrt{n}(\hat F_e-F_e)\) converges weakly to a Gaussian process \(Z(y)\) (if \(a_n^4 n\to 0\), then \(E Z(y)=0\); if \(a_n=Cn^{-1/4}\), then \(E Z(y)\not=0\)). Simulation results are presented. Reviewer: R.E.Maiboroda (Kyïv) Cited in 1 ReviewCited in 107 Documents MSC: 62G08 Nonparametric regression and quantile regression 62G20 Asymptotic properties of nonparametric inference 62J02 General nonlinear regression Keywords:heteroscedastic regression; asymptotic normality; kernel estimator PDFBibTeX XMLCite \textit{M. G. Akritas} and \textit{I. Van Keilegom}, Scand. J. Stat. 28, No. 3, 549--567 (2001; Zbl 0980.62027) Full Text: DOI