Summary
A new multivariate kernel probability density estimator is introduced and its strong uniform consistency is proved under certain regularity conditions. This result is then applied particularly to a kernel estimator whose mean vector and covariance matrix areμ n andV n, respectively, whereμ n is an unspecified estimator of the mean vector andV n, up to a multiplicative constant, the sample covariance matrix of the probability density to be estimated, respectively.
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References
Bierens HJ (1983) Sample moments integrating normal kernel estimators of multivariate density and regression functions. Sankhyā, B 45:160–192
Deheuvels P (1974) Estimation séquentielle de la densité. DSc. Thesis Univ. Paris VI
Deheuvels P (1977) Estimation non paramétrique de la densité par histogrammes généraliséś (II). Publ Inst Statist Univ Paris 22:1–23
Devroye L (1982) Bounds for the uniform deviation of empirical measures. J Mult Anal 12:72–79
Devroye L, Wagner TJ (1980) The strong uniform convergence of kernel density estimates. In: Krishnaiah PR (ed) Multivariate analysis, vol 5. North Holland, New York, pp 59–77
Gastwirth JL (1966) On robust procedures. J Amer Statist Assoc 61:929–948
Gastwirth JL, Cohen ML (1970) Small sample behavior of some robust linear estimators of location. J Amer Statist Assoc 65:946–973
Gänßler P (1983) Empirical processes. Lecture Notes — Monograph Series, vol 3, Inst Math Statist, Hayward
Huber PJ (1972) Robust statistics: a review. Ann Math Statist 43:1041–1067
Huber PJ (1981) Robust statistics. Wiley, New York
Lehmann EL (1983) Theory point estimation. Wiley, New York
Parzen E (1962) On estimation of probability density function and mode. Ann Math Statist 33:1065–1076
Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Statist 27:832–837
Shanmugam KS (1977) On a modified form of Parzen estimator for nonparametric pattern recognition 9:167–170
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Work supported by the Natural Sciences and Engineering Research Council of Canada and by the Fonds F.C.A.R. of the Province of Quebec.
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Abdous, B., Theodorescu, R. On the strong uniform consistency of a new kernel density estimator. Metrika 36, 177–194 (1989). https://doi.org/10.1007/BF02614091
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DOI: https://doi.org/10.1007/BF02614091