Summary
Consider the Parzen-Rosenblatt kernel estimate\(f_n = (1/n)\sum\limits_{i = 1}^n {K_h (x - X_i )}\) whereh>0 is a constant,K is an absolutely integrable function with integral one,K h (x)=(1/h), andX 1, ...,X n are iid random variables with common densityf onR d. We show that for all ε>0,
We also establish thatf n is relatively stable, i.e.
whenerver lim inf\(\sqrt n E\int | f_n - f| = \infty\). We also study what happens whenh is allowed to depend upon the data sequence.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Abou-Jaoude, S.: La convergenceL 1 etL ∞ de certains estimateurs d'une densite de probabilite. These de Doctorat d'Etat, University Paris VI, France, 1977
Azuma, K.: Weighted sums of certain dependent random variables. Tohoku Math. J., Ser. II.37, 357–367 (1967)
Bennett, G.: Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc.57, 33–45 (1962)
Burkholder, D.L.: Distribution function inequalities for martingales. Ann. Probab.1, 19–42 (1973)
Chow, Y.S.: Some convergence theorems for independent random variables. Ann. Math. Stat.37, 1482–1493 (1966)
Devroye, L.: The equivalence of weak, strong and complete convergence inL 1 for kernel density estimates. Ann. Stat.11, 896–904 (1983)
Devroye, L., Gyorfi, L.: Nonparametric density estimation. TheL 1 View, New York: Wiley 1985
Devroye, L.: Asymptotic performance bounds for the kernel estimate. Technical Report, School of Computer Science, McGill University, Montreal, Canada, 1987
Devroye, L.: An application of the Efron-Stein inequality in density estimation. Ann. Stat.15, 1317–1320 (1987)
Devroye, L.: An L 1 asymptotically optimal kernel estimate. Technical Report, School of Computer Science, McGill University, Montreal, Canada, 1987
Freedman, D.A.: On tail probabilities for martingales. Ann. Probab.3, 100–118 (1975)
Hall, P.: Limit theorems for stochastic measures of the accuracy of density estimators. Stochastic Processes Appl.13, 11–25 (1982)
Hall, P.: Central limit theorem for integrated square error of multivariate nonparametric density estimators. J. Multivariate Anal.14, 1–16 (1984)
Millar, P.W.: Martingales with independent increments. Ann. Math. Stat.40, 1033–1041 (1969)
Parzen, E.: On the estimation of a probability density function and the mode. Ann. Math. Stat.33, 1065–1076 (1962)
Rosenblatt, M.: Remarks on some nonparametric estimates of a density function. Ann. Math. Stat.27, 832–837 (1956)
Steiger, W.L.: A best possible Kolmogoroff-type inequality for martingales and a characteristic property. Ann. Math. Stat.40, 764–769 (1969)
Stout, W.F.: Almost sure convergence. New York: Academic Press 1974
Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge: Cambridge University Press 1927
Author information
Authors and Affiliations
Additional information
This work was sponsored by NSERC Grant A 3456 and FCAC Grant EQ-1678
Rights and permissions
About this article
Cite this article
Devroye, L. The kernel estimate is relatively stable. Probab. Th. Rel. Fields 77, 521–536 (1988). https://doi.org/10.1007/BF00959615
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00959615