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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 \(\rho\).

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