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Estimating the error distribution function in semiparametric regression. (English) Zbl 1137.62023
From the introduction: We consider the partly linear regression \(E(U-\mu(X))\) model
\[ Y= \vartheta^\top U+\rho(X)+\varepsilon, \]
where the error \(\varepsilon\) is independent of the covariate pair \((U,X)\) and the parameter \(\vartheta\) is \(k\)-dimensional. We make the following assumptions.
(F) The error \(\varepsilon\) has mean zero, a finite moment of order \(\beta>8/3\), and a density \(f\) which is Hölder with exponent \(\xi>1/3\).
(G) The distribution \(G\) of \(X\) is quasi-uniform on \([0,1]\) in the sense that \(G([0,1])=1\) and \(G\) has a density \(g\) that is bounded and bounded away from zero on \([0,1]\).
(H) The covariate vector \(U\) satisfies \(E[|U|^2]<\infty\), the matrix \(E[(U- \mu(X))(U- \mu(X)^\top))\) is positive definite, \(\mu\) is continuous and \(\tau g\) is bounded, where \(\mu(X)= E(U\,|\,X)\) and \(\tau(X)= E(|U|^2\,|\,X)\).
(R) The function \(\rho\) is twice continuously differentiable.
The goal is to estimate the distribution function \(F\) of \(\varepsilon\) based on \(n\) independent copies \((U_j, X_j, Y_j)\) of \((U, X, Y)\). Our estimator of \(F\) will be the empirical distribution function based on residuals. To obtain residuals we need estimators of \(\vartheta\) and \(\rho\).

62G08 Nonparametric regression and quantile regression
62J02 General nonlinear regression
62G20 Asymptotic properties of nonparametric inference
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