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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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