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A new method of joint nonparametric estimation of probability density and its support

Taku Moriyama

Abstract

In this paper we propose a new method of joint nonparametric estimation of probability density and its support. As is well known, nonparametric kernel density estimator has "boundary bias problem" when the support of the population density is not the whole real line. To avoid the unknown boundary effects, our estimator detects the boundary, and eliminates the boundary-bias of the estimator simultaneously. Moreover, we refer an extension to a simple multivariate case, and propose an improved estimator free from the unknown boundary bias.

A new method of joint nonparametric estimation of probability density and its support

Abstract

In this paper we propose a new method of joint nonparametric estimation of probability density and its support. As is well known, nonparametric kernel density estimator has "boundary bias problem" when the support of the population density is not the whole real line. To avoid the unknown boundary effects, our estimator detects the boundary, and eliminates the boundary-bias of the estimator simultaneously. Moreover, we refer an extension to a simple multivariate case, and propose an improved estimator free from the unknown boundary bias.

Paper Structure

This paper contains 8 sections, 4 theorems, 71 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Boundary bias reduction methods
  3. New joint estimator of probability density and its support in one dimension
  4. Asymptotic properties
  5. Simulation study in Case 1
  6. Extension to a simple multivariate case
  7. Estimating multivariate joint density with unknown but simple form of support
  8. Appendices: Some proofs

Key Result

Theorem 1

Given Assumptions 1 - 4 and that we have

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Remark 3
  • Remark 4
  • Corollary 1
  • Corollary 2
  • Remark 5
  • Remark 6
  • Theorem 2