On large bandwidth matrix values kernel smoothed estimators for multi-index models

TL;DR

This paper investigates the asymptotic properties of kernel estimators with large bandwidth matrices in multi-index models, revealing their ability to reduce the curse of dimensionality by focusing on effective dimension.

Contribution

It demonstrates that large bandwidth matrices can effectively ignore irrelevant variables, improving convergence rates based on effective rather than total dimension.

Findings

Optimal convergence depends on effective dimension, not total variables.

Kernel estimators naturally reduce curse of dimensionality.

Numerical and case studies validate theoretical results.

Abstract

The kernel smoothing with large bandwidth values causes oversmoothing or underfitting in general. However, when irrelevant variables are included, the corresponding large bandwidth values are known to have an effect of shrinking them. This study investigates asymptotic properties of the kernel conditional density estimator and the regression estimator with large bandwidth matrix elements for cases of multi-index model. It is clarified that the optimal convergence rate of the estimators depends on not the number of the variables but the effective dimension without eliminating the irrelevant variables. Thus, the kernel conditional density estimator and regression estimator are demonstrated to equip the reduction of the curse of dimensionality by nature. Finite sample performances are investigated by a numerical study, and the bandwidth selection is discussed. Finally a case study on the Boston housing data is provided.