Application of Bernstein polynomials for smooth estimation of a distribution and density function

G. Jogesh Babu, Angelo J. Canty, Yogendra P. Chaubey

Research output: Contribution to journalArticle

65 Citations (Scopus)

Abstract

The empirical distribution function is known to have optimum properties as an estimator of the underlying distribution function. However, it may not be appropriate for estimating continuous distributions because of its jump discontinuities. In this paper, we consider the application of Bernstein polynomials for approximating a bounded and continuous function and show that it can be naturally adapted for smooth estimation of a distribution function concentrated on the interval [0, 1] by a continuous approximation of the empirical distribution function. The smoothness of the approximating polynomial is further used in deriving a smooth estimator of the corresponding density. The asymptotic properties of the resulting estimators are investigated. Specifically, we obtain strong consistency and asymptotic normality under appropriate choice of the degree of the polynomial. The case of distributions with other compact and non-compact support can be dealt through transformations. Thus, this paper gives a general method for non-parametric density estimation as an alternative to the current estimators. A small numerical investigation shows that the estimator proposed here may be preferable to the popular kernel-density estimator.

Original languageEnglish (US)
Pages (from-to)377-392
Number of pages16
JournalJournal of Statistical Planning and Inference
Volume105
Issue number2
DOIs
StatePublished - Jul 1 2002

Fingerprint

Bernstein Polynomials
Density Function
Probability density function
Distribution functions
Distribution Function
Polynomials
Estimator
Empirical Distribution Function
Nonparametric Density Estimation
Kernel Density Estimator
Polynomial
Strong Consistency
Continuous Distributions
Numerical Investigation
Asymptotic Normality
Asymptotic Properties
Smoothness
Discontinuity
Continuous Function
Jump

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics

Cite this

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Application of Bernstein polynomials for smooth estimation of a distribution and density function. / Babu, G. Jogesh; Canty, Angelo J.; Chaubey, Yogendra P.

In: Journal of Statistical Planning and Inference, Vol. 105, No. 2, 01.07.2002, p. 377-392.

Research output: Contribution to journalArticle

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