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On the strong uniform consistency of a new kernel density estimator. (English) Zbl 0744.62052
The authors propose a multivariate kernel density estimator of the form \[ f_ n(x)=(n|\hbox{det} H_ n|)^{-1}\sum_{i=1}^ n K(H_ n^{-1}(x-\Phi_ n(X^*_ n,X_ i))),\qquad x\in\mathbb{R}^ k, \] where \(H_ n\) is a nonsingular \(k\times k\) matrix depending on the sample \(X^*_ n=(X_ 1,\dots,X_ n)\) and \(\Phi_ n(\cdot,\cdot)\) is a function from \(\mathbb{R}^{k\times n}\times\mathbb{R}^ k\) to \(\mathbb{R}^ k\). They prove its strong consistency (under certain regularity conditions). This result is then applied to a kernel whose mean vector and covariance matrix are \(\mu_ n\) and \(V_ n\), respectively, where \(\mu_ n\) is an unspecified estimator of the mean vector of the density \(f\) to be estimated and \(V_ n\), up to a multiplicative constant, is an estimator of the covariance matrix of \(f\).

MSC:
62G07 Density estimation
62H12 Estimation in multivariate analysis
62G20 Asymptotic properties of nonparametric inference
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[1] Bierens HJ (1983) Sample moments integrating normal kernel estimators of multivariate density and regression functions. Sankhyā, B 45:160–192 · Zbl 0534.62020
[2] Deheuvels P (1974) Estimation séquentielle de la densité. DSc. Thesis Univ. Paris VI
[3] Deheuvels P (1977) Estimation non paramétrique de la densité par histogrammes généraliséś (II). Publ Inst Statist Univ Paris 22:1–23 · Zbl 0375.62038
[4] Devroye L (1982) Bounds for the uniform deviation of empirical measures. J Mult Anal 12:72–79 · Zbl 0492.60006 · doi:10.1016/0047-259X(82)90083-5
[5] 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
[6] Gastwirth JL (1966) On robust procedures. J Amer Statist Assoc 61:929–948 · Zbl 0144.19004 · doi:10.2307/2283190
[7] Gastwirth JL, Cohen ML (1970) Small sample behavior of some robust linear estimators of location. J Amer Statist Assoc 65:946–973 · Zbl 0196.21802 · doi:10.2307/2284600
[8] Gänßler P (1983) Empirical processes. Lecture Notes – Monograph Series, vol 3, Inst Math Statist, Hayward · Zbl 1356.60003
[9] Huber PJ (1972) Robust statistics: a review. Ann Math Statist 43:1041–1067 · Zbl 0254.62023 · doi:10.1214/aoms/1177692459
[10] Huber PJ (1981) Robust statistics. Wiley, New York · Zbl 0536.62025
[11] Lehmann EL (1983) Theory point estimation. Wiley, New York · Zbl 0522.62020
[12] Parzen E (1962) On estimation of probability density function and mode. Ann Math Statist 33:1065–1076 · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[13] Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Statist 27:832–837 · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[14] Shanmugam KS (1977) On a modified form of Parzen estimator for nonparametric pattern recognition 9:167–170 · Zbl 0356.68097
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