×

zbMATH — the first resource for mathematics

Strong consistency and rates for recursive probability density estimators of stationary processes. (English) Zbl 0619.62079
Authors’s abstract: Let \(\{X_ j\}^{\infty}_{j=-\infty}\) be a vector-valued stationary process with a first-order univariate probability density f on \({\mathbb{R}}^ d\). Recursive estimation of f(x) from n not necessarily independent observations \(\{X_ j\}^ n_{j=1}\) is considered. For processes \(\{X_ j\}^{\infty}_{j=-\infty}\) which are asymptotically uncorrelated, sharp rates for the almost sure convergence of kernel-type estimators \(f_ n(x)\) are established.

MSC:
62M09 Non-Markovian processes: estimation
62G05 Nonparametric estimation
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Davis, H.I, Strong consistency of a sequential estimator of a probability density function, Bull. math. statist., 15, 49-54, (1973) · Zbl 0291.62051
[2] Deheuvels, P, Sur une famille d’estimateurs de la densité d’une variable aléatoire, C. R. acad. sci. Paris, 276, 1013-1015, (1973) · Zbl 0255.62039
[3] Györfi, L, Strong consistent density estimate from ergodic sample, J. multivariate anal., 11, 81-84, (1981) · Zbl 0449.62031
[4] Lyons, R, (), Technical Report 85T16
[5] Masry, E, Recursive probability density estimation for weakly dependent stationary processes, IEEE trans. inform. theory, IT-32, 254-267, (1986) · Zbl 0602.62028
[6] McLeish, D.L, A maximal inequality and dependent strong law, Ann. probab., 3, 829-839, (1975) · Zbl 0353.60035
[7] Stout, W.F, ()
[8] Takahata, H, Almost sure convergence of density estimators for weakly dependent stationary processes, Bull. Tokyo gakugei univ. (IV), 32, 11-32, (1980)
[9] Wegman, E; Davies, H.I, Remarks on some recursive estimators of a probability density, Ann. statist., 7, 316-327, (1979) · Zbl 0405.62031
[10] Wheeden, R.L; Zygmund, A, ()
[11] Wolverton, C.T; Wagner, T.J, Asymptotically optimal discriminant functions for pattern classification, IEEE trans. inform. theory, IT-15, 258-265, (1960) · Zbl 0172.43404
[12] Yamoto, H, Sequential estimation of a continuous probability density and mode, Bull. math. statist., 14, 1-12, (1971)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.