Summary
The class of delta-sequence estimators for a probability density includes the kernel, histogram and orthogonal series types, because each can be characterized as a collection of averages of some function that is indexed by a smoothing parameter. There are two important extensions of this class. The first allows a random smoothing parameter, for example that specified by a cross-validation method. The second allows the smoothing parameter to be a function of location, for example an estimator based on nearest-neighbor distance. In this paper a general method is presented which establishes uniform consistency for all of these estimators.
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Abramson, I.S.: Arbitrariness of the pilot estimator in adaptive kernel methods. J. Multivariate Anal. 12, 562–567 (1982a)
Abramson, I.S.: On bandwidth variation in kernel estimates — a square root law. Ann. Stat. 10, 1217–1223 (1982b)
Abramson, I.S.: Adaptive density flattening — a metric distortion principle for combating bias in nearest neighbor methods. Ann. Stat. 12, 880–886 (1984)
Bertrand-Retali, M.: Convergence uniforme stochastique d'un estimateur d'une densité de probabilité dans R. C.R. Acad. Sci. Paris, Ser. A. 278, 1449–1452 (1974)
Bleuez, J., Bosq, D.: Conditions nécessaire et suffisantes de convergence de l'estimateur de la densité par la méthode des fonctions orthogonales. C. R. Acad. Sc. Paris, Ser. A., 282:, 1023–1026 (1976)
Boneva, L., Kendall, D., Stefanov, I.: Spline transformations: Three new diagnostic aids for the statistical data analyst. J. R. Stat. Soc. B. 33, 1–70 (1971)
Breiman, L., Meisel, W., Purcell, E.: Variable kernel estimates of probability densities. Technometrics 19, 135–144 (1977)
Burman, P.: A data dependent approach to density estimation. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69, 609–628 (1985)
Devroye, L.: A note on the L 1 consistency of variable kernel estimates. Ann. Stat. 13, 1041–1049 (1985)
Devroye, L., Györfi, L.: Nonparametric density estimation: The L 1 view. New York: Wiley 1984
Devroye, L., Penrod, C.S.: The consistency of automatic kernel density estimates. Ann. Stat. 12, 1231–1249 (1984a)
Devroye, L., Penrod, C.S.: The strong uniform convergence of multivariate variable kernel estimates. Preprint 1984b
Devroye, L., Wagner, T.J.: The strong uniform consistency of kernel density estimates. In: Krishndiah, P.R. (ed.) Multivariate analysis V, pp. 59–77. New York Amsterdam: North Holland 1980
Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab. 6, 899–929 (1978)
Fix, E., Hodges, J.L.: Discriminatory analysis, nonparametric discrimination, consistency properties. Randolph Field, Texas, Project 21-49-004, Report No. 4
Földes, A., Revez, P.: A general method for density estimation. Studia Sci. Math. Hungar. 9, 81–92 (1974)
Hall, P.: Large sample optimality of least squared cross-validation in density estimation. Ann. Stat. 11, 1156–1174 (1983a)
Hall, P.: On near neighbor estimates of a multivariate density. J. Multivariate Anal. 13, 24–39 (1983b)
Hall, P.: On the rate of convergence of orthogonal series density estimators. J. R. Stat. Soc. B48, 115–122 (1986)
Hall, P.: Cross-validation and the smoothing of orthogonal series density estimators. J. Multivariate Anal. 21, 189–206 (1987)
Hall, P., Marron, J.S.: Extent to which least-squares cross-validation minimises integrated square error in nonparametric density estimation, Probab. Th. Rel. Fields 74, 567–581 (1987a)
Hall, P., Marron, J.S.: On the amount of noise inherent in bandwidth selection for a kernel density estimator. Ann. Stat. 15, 163–181 (1987b)
Hall, P., Marron, J.S.: Variable window width kernel estimates of probability densities. Probab. Th. Rel. Fields (Ms. 372)
Hall, P., Shucany, W.: A local cross-validation algorithm. Preprint 1988
Kim, B.K., Van Ryzin, J.: Uniform consistency of a histogram density estimator and model estimation. Comm. Stat. 4, 303–315 (1975)
Krieger, A.M., Pickands, J. III: Weak convergence and efficient density estimation at a point. Ann. Statist. 9, 1066–1078 (1981)
Loftsgaarden, D.O., Quesenberry, C.P.: A nonparametric estimate of a multivariate density function. Ann. Math. Stat. 36, 1049–1051 (1965)
Mack, Y.P., Rosenblatt, M.: Multivariate k-nearest neighbor density estimates. J. Multivariate Anal. 9, 1–15 (1979)
Marron, J.S.: An asymptotically efficient solution to the bandwidth problem of kernel density estimation. Ann. Stat. 13, 1011–1023 (1985)
Marron, J.S.: A comparison of cross-validation techniques in density estimation. Ann. Stat. 15, 152–162 (1987)
Mielniczuk, J., Sarda, P., Vieu, P.: Local data-driven bandwidth choice for density estimation. Preprint 1988
Muller, H.G., Stadtmuller, U.: Variable bandwidth kernel estimates of regression curves. Ann. Stat. 15, 182–201 (1987)
Nolan, D.: Uniform convergence of variable kernel estimates. Unpublished prospectus. Yale University 1984
Nolan, D., Pollard, D.: U-processes: rates of convergence. Ann. Stat. 15, 780–789 (1987)
Pollard, D.: Convergence of stochastic processes. New York Berlin Heidelberg: Springer 1984
Pollard, D.: Rates of uniform almost-sure convergence for empirical processes indexed by unbounded classes of functions. Preprint 1987
Prakasa Rao, B.L.S.: Nonparametric functional estimation. New York: Academic Press 1983
Revesz, P.: On empirical density function. Period. Math. Hungar. 2, 85–110 (1972)
Rudemo, M.: Empirical choice of histograms and kernel density estimators. Scand. J. Stat. 9, 65–78 (1982)
Scott, D.W.: Frequency Polygons: theory and application, J. Am. Stat. Soc. 80, 348–354 (1985a)
Scott, D.W.: Averaged shifted histograms: effective nonparametric density estimators in several dimensions. Ann. Stat. 13, 1024–1040 (1985b)
Silverman, B.W.: Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Ann. Stat. 6, 177–184 (1978)
Silverman, B.W.: Density estimation for statistics and data analysis. New York: Chapman and Hall 1986
Stone, C.J.: An asymptotically optimal window selection rule for kernel density estimates. Ann. Stat. 12, 1285–1297 (1984)
Stone, C.J.: An asymptotically optimal histogram selection rule. InL LeCam, L., Olshen, R.A. (eds.) Conference in honor of Jerzy Neyman and Jack Kiefer. Proceedings, Berkeley 1985. Vol. II, pp. 513–520. Wadsworth 1985
Tapia, R.A., Thompson, J.R.: Nonparametric probability density estimation. Baltimore: Johns Hopkins University Press 1978
Van Ryzin, J.: A histogram method of density estimation. Comm. Stat. 2, 493–506 (1973)
Vieu, P.: Nonpaprametric regression: Optimal global bandwidth. Preprint (1988)
Wahba, G.: A polynomial algorithm for density estimation. Ann. Math. Stat. 42, 1870–1886 (1971)
Wahba, G.: Interpolating spline methods for density estimation I. Equi-spaced knots. Ann. Stat. 3, 30–48 (1975)
Wahba, G.: Data-based optimal smoothing of orthogonal series density estimates. Ann. Stat. 9, 146–156 (1981)
Walter, G.G., Blum, J.: Probability density estimation using delta sequences. Ann. Stat. 7, 328–340 (1979)
Woodroofe, M.: On choosing a delta sequence. Ann. Math. Stat. 41, 1665–1671 (1970)
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Research partially supported by AFOSR Grant No. S-49620-82-C-0144, and by NSF Grant DMS-850-3347
Research supported by NSF Grant DMS-8400602
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Nolan, D., Marron, J.S. Uniform consistency of automatic and location-adaptive delta-sequence estimators. Probab. Th. Rel. Fields 80, 619–632 (1989). https://doi.org/10.1007/BF00318909
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DOI: https://doi.org/10.1007/BF00318909