Abstract
It is a known fact that some estimators of smooth distribution functions can outperform the empirical distribution function in terms of asymptotic (integrated) mean-squared error. In this paper, we show that this is also true of Bernstein polynomial estimators of distribution functions associated with densities that are supported on a closed interval. Specifically, we introduce a higher order expansion for the asymptotic (integrated) mean-squared error of Bernstein estimators of distribution functions and examine the relative deficiency of the empirical distribution function with respect to these estimators. Finally, we also establish the (pointwise) asymptotic normality of these estimators and show that they have highly advantageous boundary properties, including the absence of boundary bias.
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References
Aggarwal O.P. (1995) Some minimax invariant procedures for estimating a cumulative distribution function. Annals of Mathematical Statistics 26: 450–463
Altman N., Léger C. (1995) Bandwidth selection for kernel distribution function estimation. Journal of Statistical Planning and Inference 46: 195–214
Azzalini A. (1981) A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68: 326–328
Babu G.J., Chaubey Y.P. (2006) Smooth estimation of a distribution function and density function on a hypercube using Bernstein polynomials for dependent random vectors. Statistics and Probability Letters 76: 959–969
Babu G.J., Canty A.J., Chaubey Y.P. (2002) Application of Bernstein polynomials for smooth estimation of a distribution and density function. Journal of Statistical Planning and Inference 105: 377–392
Billingsley P. (1995) Probability and measure. (3rd edn.). Wiley, New York
Bowman A., Hall P., Prvan T. (1998) Bandwidth selection for the smoothing of distribution functions. Biometrika 85: 799–808
Brown B M., Chen S.X. (1999) Beta-Bernstein smoothing for regression curves with compact support. Scandinavian Journal of Statistics 26: 47–59
Chacón J.E., Rodríguez-Casal A. (2010) A note on the universal consistency of the kernel distribution function estimator. Statistics and Probability Letters 80: 1414–1419
Chang I.-S., Hsiung C.A., Wu Y.-J., Yang C.-C. (2005) Bayesian survival analysis using Bernstein polynomials. Scandinavian Journal of Statistics 32: 447–466
Choudhuri N., Ghosal S., Roy A. (2004) Bayesian estimation of the spectral density of a time series. Journal of the American Statistical Association 99: 1050–1059
Cressie N. (1978) A finely tuned continuity correction. Annals of the Institute of Statistical Mathematics 30: 435–442
Falk M. (1983) Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Statistica Neerlandica 37: 73–83
Ghosal S. (2001) Convergence rates for density estimation with Bernstein polynomials. Annals of Statistics 29: 1264–1280
Hjort L.H., Walker S.G. (2001) A note on kernel density estimators with optimal bandwidths. Satistics and Probability Letters 54: 153–159
Hodges J.L. Jr., Lehman E.L. (1970) Deficiency. Annals of Mathematical Statistics 41: 783–801
Jones M.C. (1990) The performance of kernel density functions in kernel distribution function estimation. Satistics and Probability Letters 9: 129–132
Kakizawa Y. (2004) Bernstein polynomial probability density estimation. Journal of Nonparametric Statistics 16: 709–729
Leblanc A. (2009) Chung–Smirnov property for Bernstein estimators of distribution functions. Journal of Nonparametric Statistics 21: 133–142
Leblanc A. (2010) A bias-reduced approach to density estimation using Bernstein polynomials. Journal of Nonparametric Statistics 22: 459–475
Liu R., Yang L. (2008) Kernel estimation of multivariate cumulative distribution function. Journal of Nonparametric Statistics 20: 661–677
Lorentz G.G. (1986) Bernstein polynomials. (2nd edn.). Chelsea Publishing, New York
Petrone S. (1999) Bayesian density estimation using Bernstein polynomials. Canadian Journal of Statistics 27: 105–126
Petrone S., Wasserman L. (2002) Consistency of Bernstein polynomial posteriors. Journal of the Royal Statistical Society, Series B 64: 79–100
Rao B.L.S.P. (2005) Estimation of distribution and density functions by generalized Bernstein polynomials. Indian Journal of Pure and Applied Mathematics 36: 63–88
Read R.R. (1972) Asymptotic inadmissibility of the sample distribution function. Annals of Mathematical Statistics 43: 89–95
Reiss R.-D. (1981) Nonparametric estimation of smooth distribution functions. Scandinavian Journal of Statistics 8: 116–119
Serfling R. J. (1980). Approximation theorems of mathematical statistics. New York, Wiley
Silverman B.W. (1986) Density estimation. Boca Raton, Chapman & Hall/CRC
Swanepoel J.W.H., Van Graan F.C. (2005) A new kernel distribution function estimator based on a non-parametric transformation of the data. Scandinavian Journal of Statistics 32: 551–562
Tenbusch A. (1994) Two-dimensional Bernstein polynomial density estimators. Metrika 41: 233–253
Tenbusch A. (1997) Nonparametric curve estimation with Bernstein estimates. Metrika 45: 1–30
Vitale R.A. (1975) A Bernstein polynomial approach to density function estimation. Statistical Inference and Related Topics 2: 87–99
Watson G.S., Leadbetter M.R. (1964) Hazard analysis II. Sankhya A 26: 101–116
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Leblanc, A. On estimating distribution functions using Bernstein polynomials. Ann Inst Stat Math 64, 919–943 (2012). https://doi.org/10.1007/s10463-011-0339-4
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DOI: https://doi.org/10.1007/s10463-011-0339-4