Commenced in January 2007
Paper Count: 30184
Orthogonal Polynomial Density Estimates: Alternative Representation and Degree Selection
Abstract:The density estimates considered in this paper comprise a base density and an adjustment component consisting of a linear combination of orthogonal polynomials. It is shown that, in the context of density approximation, the coefficients of the linear combination can be determined either from a moment-matching technique or a weighted least-squares approach. A kernel representation of the corresponding density estimates is obtained. Additionally, two refinements of the Kronmal-Tarter stopping criterion are proposed for determining the degree of the polynomial adjustment. By way of illustration, the density estimation methodology advocated herein is applied to two data sets.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1329791Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1860
 Izenman A J. Recent development in nonparametric density density estimation. Journal of the American Statistical Association, 1991, 86:205- 224.
 Watson G S. Density estimation by orthogonal series. Annals of Mathematical Statistics, 1969, 40:1496-1498.
 Rosenblatt M. Curve estimates. The Annals of Mathematical Statistics, 1971, 42:1815-1842.
 Hall P. On the rate of convergence of orthogonal series density estimators. Journal of Royal the Royal Statistical Society, Ser. B, 1986, 48:115-122.
 Johnstone I M, Silverman B W. Speed of estimation in positron emission tomography and related inverse problems. The Annals of Statistics, 1990, 18:251-280.
 Provost S B. Moment-based density approximants. The Mathematica Journal, 2005, 9:727-756.
 Ha H-T, Provost S B. A viable alternative to resorting to statistical tables. Communications in Statistics-Simulation and Computation, 2007, 36:1135-1151.
 Provost S B, Ha H-T. On the inversion of certain moment matrices. Linear Algebra and Its Applications, 2009, 430:2650-2658.
 Brunk H B. Univariate density estimation by orthogonal series. Biometrika, 1978, 65:521-528.
 Parzen E. Nonparametric statistical data modeling. Journal of the American Statistical Association, 1979, 74:105-121.
 Ruppert D, Cline D B H. Bias reduction in kernel density estimation by smoothed empirical transformations. Institute of Mathematical Statistics, 1994, 22:185-210.
 Hjort N L, Jones M C. Locally parametric nonparametric density estimation. The Annals of Statistics, 1996, 24:1619-1647.
 Rao C R. Linear Statistical Inference and Its Applications. New York: Wiley, 1973.
 Arfken G. Gram-Schmidt Orthogonalization. Orlando: Academic Press, 1985.
 Anderson G L, De Figueiredo R J P. An adaptive orthogonal-series estimator for probability density functions. The Annals of Statistics, 1980, 8:347-376.
 Alexits G. Convergence Problems of Orthogonal Series. New York: Pergamon Press, 1961.
 Hildebrand F B. Introduction to Numerical Analysis. New York: McGraw-Hill, 1956.
 Kronmal R, Tarter M. The estimation of probability densities and cumulatives by Fourier series methods. Journal of the American Statistical Association, 1968, 63:925-952.
 Diggle P J, Hall P. The selection of terms in an orthogonal series density estimator. Journal of American Statistical Association, 1986, 81:230-233.
 Tarter M, Kronmal R. An introduction to the implementation and theory of nonparametric density estimation. The American Statistician, 1976, 30:105-112.
 Simonoff J S. Smoothing Methods in Statistics. New York: Springer, 1996.
 Roeder K. Density estimation with confidence sets exemplified by superclusters and voids in galaxies. Journal of the American Statistical Association, 1990, 85:617-624.