Abstract
Let X 1, X 2, ..., X n be independent observations from an (unknown) absolutely continuous univariate distribution with density f and let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GabmOzayaajaGaaiikaiaadIhacaGGPaGaeyypa0Jaaiikaiaad6ga% caWGObGaaiykamaaCaaaleqabaGaeyOeI0IaaGymaaaakmaaqadaba% Gaam4saiaacUfadaWcgaqaaiaacIcacaWG4bGaeyOeI0Iaamiwamaa% BaaaleaacaWGPbaabeaakiaacMcaaeaacaWGObGaaiyxaaaaaSqaai% aadMgacqGH9aqpcaaIXaaabaGaamOBaaqdcqGHris5aaaa!5356!\[\hat f(x) = (nh)^{ - 1} \sum\nolimits_{i = 1}^n {K[{{(x - X_i )} \mathord{\left/ {\vphantom {{(x - X_i )} {h]}}} \right. \kern-\nulldelimiterspace} {h]}}} \] be a kernel estimator of f(x) at the point x, \s-∞<x<∞, with h=h n (h n →O and nh n →∞, as n→∞) the bandwidth and K a kernel function of order r. “Optimal” rates of convergence to zero for the bias and mean square error of such estimators have been studied and established by several authors under varying conditions on K and f. These conditions, however, have invariably included the assumption of existence of the r-th order derivative for f at the point x. It is shown in this paper that these rates of convergence remain valid without any differentiability assumptions on f at x. Instead some simple regularity conditions are imposed on the density f at the point of interest. Our methods are based on certain results in the theory of semi-groups of linear operators and the notions and relations of calculus of “finite differences”.
Similar content being viewed by others
References
Bartlett, M. S. (1963). Statistical estimation of density functions, Sankhyā Ser. A, 25, 245–254.
Butzer, P. L. and Berens, H. (1967). Semi-Groups of Operators and Approximation, Springer, New York.
Davis, K. B. (1975). Mean square error properties of density estimates, Ann. Statist., 3, 1025–1030.
Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: L 1-view, Wiley, New York.
Farrell, R. H. (1972). On best obtainable asymptotic rates of convergence in estimation of a density function at a point, Ann. Math. Statist., 43, 170–180.
Hardy, G. H. (1955). A Course of Pure Mathematics, The University Press, Cambridge.
Hewitt, E. and Stromberg, K. (1965). Real and Abstract Analysis, Springer, New York.
Hille, E. (1944). Functional Analysis and Semi-groups, American Mathematical Society Publication, New York.
Ibiragimov, I. A. and Has'minski, R. Z. (1981). Statistical Estimation, Springer, New York.
Johns, M. V. and Van Ryzin, J. (1972). Convergence rates for empirical Bayes two-action problems II. Continuous case, Ann. Math. Statist., 43, 934–947.
Karunamuni, R. J. and Mehra, K. L. (1990). Improvements on strong uniform consistency of some known kernel estimates of a density and its derivatives, Statist. Probab. Lett., 9, 133–140.
Lehmann, E. L. (1983). Theory of Point Estimation, Wiley, New York.
Loh, W.-Y. (1984). Bounds on ARE's for restricted classes of distributions defined via tailordering, Ann. Statist., 12, 685–701.
Müller, H. G. (1984). Smooth kernel estimators of densities, regression curves and modes, Ann. Statist., 12, 766–774.
Müller, H. G. and Gasser, T. (1979). Optimal convergence properties of kernel estimates of derivatives of densities, Smoothing Techniques for Curve Estimation (Proceedings) (eds. T. Gasser and M. Rosenblatt), 144–154, Springer, New York.
Parzen, E. (1962). On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065–1076.
Rosenblatt, M. (1971). Curve estimates, Ann. Math. Statist., 42, 1815–1842.
Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman and Hall, New York.
Singh, R. S. (1974). Estimation of derivatives of average of densities and sequence-compound estimation in exponential families, Research Memo. RM-18, Dept. of Statistics and Probability, Michigan State University.
Singh, R. S. (1977). Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density, Ann. Statist., 5, 394–399.
Singh, R. S. (1978). Nonparametric estimation of derivatives of average of μ-densities with covergence rates and applications, SIAM J. Appl. Math., 35, 637–649.
Singh, R. S. (1979). Empirical Bayes estimation in Lebesgue-exponential families with rates near the best possible rate, Ann. Statist., 7, 890–902.
Tapia, R. A. and Thompson, J. R. (1978). Nonparametric Probability Density Estimation, Johns Hopkins University Press, Baltimore, Maryland.
Wahba, G. (1975). Optimal convergence properties of variable knot, kernel and orthogonal-series methods for density estimation, Ann. Statist., 3, 15–29.
Yamato, H. (1972). Some statistical properties of estimators of density and distribution functions, Bull. Math. Statist., 19, 113–131.
Author information
Authors and Affiliations
Additional information
This research was supported in part by grants from the Natural Sciences and Engineering Research Council of Canada and the University of Alberta Central Research Fund.
About this article
Cite this article
Karunamuni, R.J., Mehra, K.L. Optimal convergence properties of kernel density estimators without differentiability conditions. Ann Inst Stat Math 43, 327–346 (1991). https://doi.org/10.1007/BF00118639
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00118639