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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.

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