Contribution to the Nonparametric Estimation of the Density of the Regression Errors (Doctoral Thesis)

TL;DR

This thesis develops nonparametric methods to estimate the density of regression errors without observing them directly, addressing the curse of dimensionality by using residuals and conditional density estimators.

Contribution

It proposes two novel approaches to estimate the error density nonparametrically, avoiding the curse of dimensionality and analyzing their convergence rates.

Findings

Proposed estimators achieve favorable convergence rates.

Evaluated the impact of residual estimation on density accuracy.

Characterized optimal bandwidth choices for the estimators.

Abstract

This thesis deals with the nonparametric estimation of density f of the regression error term E of the model Y=m(X)+E, assuming its independence with the covariate X. The difficulty linked to this study is the fact that the regression error E is not observed. In a such setup, it would be unwise, for estimating f, to use a conditional approach based upon the probability distribution function of Y given X. Indeed, this approach is affected by the curse of dimensionality, so that the resulting estimator of the residual term E would have considerably a slow rate of convergence if the dimension of X is very high. Two approaches are proposed in this thesis to avoid the curse of dimensionality. The first approach uses the estimated residuals, while the second integrates a nonparametric conditional density estimator of Y given X. If proceeding so can circumvent the curse of dimensionality, a challenging issue is to evaluate the impact of the estimated residuals on the final estimator of the density f. We will also attempt to determine the pointwise rate of convergence of our proposed estimators. One our main aims is to characterize the optimal choices of the first and second step bandwidths used for estimating m and f respectively.