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Estimation of a two-dimensional distribution function under association. (English) Zbl 1031.62027
Summary: By considering an associated and strictly stationary sequence of random variables, say \(X_n\), \(n\geqslant 1\), we study the properties of a histogram estimator for the two-dimensional distribution function of (\(X_1,X_{k+1}\)). We find conditions on the covariance structure of the original random variables for the almost sure convergence of the estimator and for the convergence in distribution of the finite-dimensional distributions. We also characterize the mean square error (MSE) and find its convergence rate, under assumptions on the convergence rate of the covariances.

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G30 Order statistics; empirical distribution functions
Full Text: DOI
[1] Anderson, T.W.; Darling, D.A., Asymptotic theory of certain goodness of fit criteria based on stochastic processes, Ann. math. statist., 23, 193-212, (1952) · Zbl 0048.11301
[2] Billingsley, P., Convergence of probability measures, (1968), Wiley New York · Zbl 0172.21201
[3] Cai, Z.; Roussas, G.G., Efficient estimation of a distribution function under quadrant dependence, Scand. J. statist., 25, 211-224, (1998) · Zbl 0904.62045
[4] Donsker, M.D., 1951. An invariance principle for certain probability limit theorems. Mem. Amer. Math. Soc., no. 6, 12pp. · Zbl 0042.37602
[5] Doukhan, P.; Massart, P.; Rio, E., Invariance principles for the empirical measure of a weakly dependent process, Ann. inst. H. Poincaré, 31, 393-427, (1995) · Zbl 0817.60028
[6] Esary, J.D.; Proschan, F.; Walkup, D.W., Association of random variables, with applications, Ann. math. statist., 38, 1466-1474, (1967) · Zbl 0183.21502
[7] Lebowitz, J., Bounds on the correlations and analycity properties of ferromagnetic Ising spin systems, Comm. math. phys., 28, 313-321, (1972)
[8] Newman, C.M., Normal fluctuations and the FKG inequalities, Comm. math phys., 74, 119-128, (1980) · Zbl 0429.60096
[9] Newman, C.M., 1984. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Inequalities in Statistics and Probability, IMS Lecture Notes, Monograph Series, Vol. 5, Inst. Math. Statist., Hayward, CA, pp. 127-140.
[10] Oliveira, P.E.; Suquet, C., L2[0,1] weak convergence of the empirical process for dependent variables, (), 331-344 · Zbl 0831.60037
[11] Oliveira, P.E.; Suquet, C., Weak convergence in Lp[0,1] of the uniform empirical process under dependence, Statist. probab. lett., 39, 363-370, (1998) · Zbl 0923.60004
[12] Roussas, G.G., Asymptotic normality of a smooth estimate of a random field distribution function under association, Statist. probab. lett., 24, 77-90, (1995) · Zbl 0830.62040
[13] Sadivoka, S.M., Two-dimensional analogies of an inequality of Esseen with applications to the central limit theorem, Theory probab. appl., 11, 325-335, (1996)
[14] Sen, P.K., A note on weak convergence of empirical processes for sequences of φ-mixing random variables, Ann. math. statist., 42, 2131-2133, (1971) · Zbl 0226.60008
[15] Shao, Q.M., Weak convergence of multidimensional weighted empirical processes for strong mixing sequences, Chinese ann. math. ser., A 7, 547-552, (1986) · Zbl 0652.60032
[16] Shao, Q.M.; Yu, H., Weak convergence for weighted empirical process of dependent sequences, Ann. probab., 24, 2098-2127, (1996) · Zbl 0874.60006
[17] Yoshihara, K., Billingsley’s theorems on empirical processes of strong mixing sequences, Yokohama math. J., 23, 1-7, (1975) · Zbl 0339.60030
[18] Yu, H., A glivenko – cantelli lemma and weak convergence for empirical processes of associated sequences, Probab. theory relat. fields, 95, 357-370, (1993) · Zbl 0792.60018
[19] Watson, G.S., Goodness-of-fit tests on a circle, Biometrika, 48, 109-114, (1961) · Zbl 0212.21905
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