zbMATH — the first resource for mathematics

Convergence rates for the estimation of two-dimensional distribution functions under association and estimation of the covariance of the limit empirical process. (English) Zbl 1092.62054
Summary: Let \(X_n,n\geq 1\), be an associated and strictly stationary sequence of random variables, having marginal distribution function \(F\). The limit in distribution of the empirical process, when it exists, is a centred Gaussian process with covariance function depending on terms of the form \[ \varphi_k(s,t)=P(X_1\leq s, X_{k+1}\leq t)-F(s)F(t). \] We prove the almost sure consistency for the histogram to estimate each \(\varphi_k\) and also to estimate the covariance function of the limit empirical process, identifying, for both, uniform almost sure convergence rates. The convergence rates depend on a suitable version of an exponential inequality. The rates obtained, assuming the covariances to decrease geometrically, are of order \(n^{-1/3}\log^{2/3}n\) for the estimator of \(\varphi_k\) and of order \(n^{-1/3}\log^{5/3}n\) for the estimator of the covariance function.

62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
60F15 Strong limit theorems
62G05 Nonparametric estimation
Full Text: DOI
[1] Shorack G. R., Empirical Processes with Applications to Statistics (1986) · Zbl 1170.62365
[2] DOI: 10.1214/aoms/1177698701 · Zbl 0183.21502 · doi:10.1214/aoms/1177698701
[3] Newman C., Inequalities in Statistics and Probability, IMS Lecture Notes–Monograph Series 5 pp 127– (1984)
[4] DOI: 10.1016/S0246-0203(00)00140-0 · Zbl 0968.60019 · doi:10.1016/S0246-0203(00)00140-0
[5] Oliveira P. E., Lecture Notes in Statistics 103, Wavelets and Statistics. Actes des XV Rencotres Franco-Belges de Statisticiens (Ondelettes et Statistique) pp 331– (1995)
[6] DOI: 10.1016/S0167-7152(98)00091-1 · Zbl 0923.60004 · doi:10.1016/S0167-7152(98)00091-1
[7] DOI: 10.1016/S0378-3758(01)00296-8 · Zbl 1031.62027 · doi:10.1016/S0378-3758(01)00296-8
[8] DOI: 10.1016/S0167-7152(98)00240-5 · Zbl 0955.60018 · doi:10.1016/S0167-7152(98)00240-5
[9] DOI: 10.1080/10485259908832769 · Zbl 0979.62017 · doi:10.1080/10485259908832769
[10] DOI: 10.1137/1111035 · doi:10.1137/1111035
[11] DOI: 10.1007/BF01197754 · Zbl 0429.60096 · doi:10.1007/BF01197754
[12] DOI: 10.1016/0167-7152(94)00151-W · Zbl 0830.62040 · doi:10.1016/0167-7152(94)00151-W
[13] DOI: 10.1007/BF01645632 · doi:10.1007/BF01645632
[14] Devroye L., Nonparametric Functional Estimation and Related Topics pp 31– (1991) · doi:10.1007/978-94-011-3222-0_3
[15] Loéve, M. 1977.Probability Theory I, 160New York: Springer-Verlag.
[16] Loéve, M. 1977.Probability Theory I, 240New York: Springer-Verlag.
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.