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

Taku Moriyama

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.

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

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.
Paper Structure (13 sections, 10 theorems, 85 equations, 1 table)

This paper contains 13 sections, 10 theorems, 85 equations, 1 table.

Table of Contents

  1. Introduction
  2. Case of independence in regression or conditional density estimation
  3. Density estimator with large bandwidth matrix
  4. Case of independence in regression estimation
  5. Case of independence in conditional density estimation
  6. Case of conditional independence in regression or conditional density estimation
  7. Case of conditional independence in regression estimation
  8. Case of conditional independence in conditional density estimation
  9. Case of multi-index model in regression or conditional density estimation
  10. Case of multi-index model in regression
  11. Case of multi-index model in conditional density estimation
  12. Simulation and Case study
  13. Conclusion

Key Result

Lemma 1

Suppose (i) $h \to \infty$ as $n \to \infty$, (ii) $k(\bm{0})$ exists, (iii) $k$ is two times continuously differentiable around the origin, (iv) $\nabla (k)(\bm{0}) = \bm{0}$ and (v) $2$th moment of $f$ exist. Then,

Theorems & Definitions (27)

  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 1
  • proof
  • ...and 17 more