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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
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