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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.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G30 Order statistics; empirical distribution functions 60F15 Strong limit theorems 62G05 Nonparametric estimation
##### Keywords:
histogram estimator; stationarity
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##### References:
 [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.
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