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Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. (English) Zbl 0861.62029
Summary: This paper presents a method for estimating the model \(\Lambda(Y)= \beta'X+U\), where \(Y\) is a scalar, \(\Lambda\) is an unknown increasing function, \(X\) is a vector of explanatory variables, \(\beta\) is a vector of unknown parameters, and \(U\) has unknown cumulative distribution function \(F\). It is not assumed that \(\Lambda\) and \(F\) belong to known parametric families; they are estimated nonparametrically. This model generalizes a large number of widely used models that make stronger a priori assumptions about \(\Lambda\) and/or \(F\).
The paper develops \(n^{1/2}\)-consistent, asymptotically normal estimators of \(\Lambda\), \(F\), and quantiles of the conditional distribution of \(Y\). Estimators of \(\beta\) that are \(n^{1/2}\)-consistent and asymptotically normal already exist. The results of Monte Carlo experiments indicate that the new estimators work reasonably well in samples of size 100.

62G07 Density estimation
62P20 Applications of statistics to economics
62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
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