×

Almost sure \(L_ 1\)-norm convergence for data-based histogram density estimates. (English) Zbl 0614.62044

Let F be a one-dimensional distribution function with density f, and let \(X_ 1,...,X_ n\) denote an i.i.d. sample drawn from F. It is shown that two conditions of a general nature on the random grids imply strong \(L_ 1\)-consistency of the data-based histogram density estimator \(f_ n(x)=f_ n(x;X_ 1,...,X_ n)\), i.e. \[ \int^{\infty}_{- \infty}| f_ n(x)-f(x)| dx\to 0\quad a.s.\quad as\quad n\to \infty. \] The generalization of the result to the multidimensional case is also indicated.
Reviewer: E.Häusler

MSC:

62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
60F25 \(L^p\)-limit theorems
60F15 Strong limit theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alexander, K. S., Probability inequalities for empirical processes and a law of the iterated logarithm, Ann. Probab., 12, 1041-1067 (1984) · Zbl 0549.60024
[2] Chen, J.; Rubin, H., On the consistency of the data-based histogram density estimator, (Technical Report, No. 84-11 (1984), Purdue University: Purdue University Lafayette, Ind)
[3] Chow, Y. S.; Geman, S.; Wu, L. D., Consistent cross-validated density estimates, Ann. Statist., 11, 25-38 (1983) · Zbl 0509.62033
[4] Csörgö, M.; Révész, P., (Strong Approximations in Probability and Statistics (1981), Akadémiai Kiadó: Akadémiai Kiadó Budapest) · Zbl 0539.60029
[5] Devroye, L. P.; Wagner, T. J., The strong uniform consistency of kernel density estimates, (Krishnaiah, P., Multivariate Analysis V (1980), North-Holland: North-Holland Amsterdam), 59-77 · Zbl 0431.62024
[6] Komlós, J.; Major, P.; Tusnády, G., An approximation of partial sums of independent R.V.’s and the sample DF I, Z. Wahrsch. Verw. Gebiete, 32, 111-131 (1975) · Zbl 0308.60029
[7] Vapnik, V. N.; Chervonenkis, A. Ya., On uniform convergence of the frequecies of events to their probabilities, Theor. Probab. Appl., 16, 264-280 (1971) · Zbl 0247.60005
[8] Wang, Shou-ren; Chen, Xi-ru, \(L_1\)-norm strong consistency for data-based histogram estimates of multidimensional density (1985), Personal communication
[9] Zhao, L. C., An inequality concerning the deviation between theoretical and empirical distributions, (Technical Report, No. 85-30 (1985), Center for Multivariate Analysis, University of Pittsburgh) · Zbl 0667.62029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.