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Exponential inequality for associated random variables. (English) Zbl 0955.60018
Summary: Under mild conditions, a Bernstein-Hoeffding-type inequality is established for covariance invariant positively associated random variables. A condition is given for almost sure convergence, and the associated rate of convergence is specified in terms of the underlying covariance function.

##### MSC:
 6e+16 Inequalities; stochastic orderings
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##### References:
 [1] Bagai, I.; Prakasa Rao, B.L.S., Estimation of the survival function for stationary associated processes, Statist. probab. lett., 12, 385-391, (1991) · Zbl 0749.62057 [2] Bagai, I.; Prakasa Rao, B.L.S., Kernel-type density and failure rate estimation for associated sequences, Ann. inst. statist. math., 47, 253-266, (1995) · Zbl 0833.62036 [3] Barlow, R.E., Proschan, F., 1975. Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York. · Zbl 0379.62080 [4] Birkel, T., Moment bounds for associated sequences, Ann. probab., 16, 1184-1193, (1988) · Zbl 0647.60039 [5] Birkel, T., On the convergence rate in the central limit theorem for associated processes, Ann. probab., 16, 1685-1698, (1988) · Zbl 0658.60039 [6] Birkel, T., A note on the strong law of large numbers for positively dependent random variables, Statist. probab. lett., 7, 17-20, (1989) · Zbl 0661.60048 [7] Cai, Z.W.; Roussas, G.G., Efficient estimation of a distribution function under quadrant dependence, Scand. J. statist., 24, 1-14, (1997) [8] Cai, Z.W.; Roussas, G.G., Smooth estimate of quantiles under association, Statist. probab. lett., 36, 275-287, (1997) · Zbl 0946.62039 [9] Cai, Z.W., Roussas, G.G., 1998a. Berry-Esseen bounds for smooth estimator of a distribution function under association. J. Nonparametric Statist., to appear. · Zbl 0980.62040 [10] Cai, Z.W., Roussas, G.G., 1998b. Kaplan-Meier estimator under association. J. Multivariate Anal., to appear. · Zbl 1030.62521 [11] Cohen, A., Sackrowitz, H.B., 1992. Some remarks on a notion of positive dependence, association, and unbiased testing. In: Shaked, M., Tong, Y.L. (Eds.), Stochastic Inequalities. IMS, Lecture Notes-Monograph Series, vol. 22. · Zbl 1402.62027 [12] Cohen, A., Sackrowitz, H.B., 1994. Association and unbiased tests in statistics. In: Shaked, M., Shanthikumar, J.G. (Eds.), Stochastic Orders and Their Applications. Academic Press, Boston. [13] Cox, J.T.; Grimmett, G., Central limit theorem for associated random variables and the percolation model, Ann. probab., 12, 514-528, (1984) · Zbl 0536.60094 [14] Devroye, L., 1991. Exponential inequalities in nonparametric estimation. In: Roussas, G. (Ed.), Nonparametric Functional Estimation and Related Topics. Kluwer Academic Publishers, Dordrecht, pp. 31-44. · Zbl 0739.62025 [15] Esary, J.D.; Proschan, F.; Walkup, D.W., Association of random variables, with applications, Ann. math. statist., 38, 1466-1474, (1967) · Zbl 0183.21502 [16] Fortuin, C.M.; Kasteleyn, P.W.; Ginibre, J., Correlation inequalities on some partially ordered sets, Commun. math. phys., 22, 89-103, (1971) · Zbl 0346.06011 [17] Harris, T.E., A lower bound for the critical probability in a certain percolation process, Proc. camb. phil. soc., 56, 13-20, (1960) · Zbl 0122.36403 [18] Joag-Dev, K.; Proschan, F., Negative association of random variables, with applications, Ann. statist., 11, 286-295, (1983) · Zbl 0508.62041 [19] Kemperman, J.H.B., 1977. On the FKG-inequality for measures on a partially ordered space. Nederl. kad. Wetensch. Indag. Math. Proc. Ser. A80(4), 313-331. · Zbl 0384.28012 [20] Lebowitz, J.L., Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems, Commun. math. phys., 28, 313-321, (1972) [21] Newman, C.M., Normal fluctuations and the FKG inequalities, Commun. math. phys., 74, 119-128, (1980) · Zbl 0429.60096 [22] Newman, C.M., 1984. Asymptotic independence and limit theorems for positively and negatively dependent random variables. In: Tong, Y.L. (Ed.), Inequalities in Statistics and Probability. IMS Lecture Notes-Monograph Series, vol. 5, pp. 127-140, Hayward, CA. [23] Newman, C.M., 1990. Ising models and dependent percolation. In: Block, H.W., Sampson, A.R., Savits, T.H. (Eds.), Topics in Statistical Dependence. IMS Lecture Notes-Monograph Series, vol. 16, pp. 395-401. [24] Newman, C.M.; Wright, A.L., An invariance principle for certain dependent sequences, Ann. probab., 9, 671-675, (1981) · Zbl 0465.60009 [25] Newman, C.M.; Wright, A.L., Associated random variables and martingale inequalities, Z. wahrsch. verw. gebiete, 59, 361-371, (1982) · Zbl 0465.60010 [26] Preston, C.J., A generalization of the FKG inequalities, Commun. math. phys., 36, 233-241, (1974) [27] Roussas, G.G., Kernel estimates under association: strong uniform consistency, Statist. probab. lett., 12, 393-403, (1991) · Zbl 0746.62045 [28] Roussas, G.G., Curve estimation in random fields of associated processes, J. nonparametric statist., 2, 215-224, (1993) · Zbl 1360.62479 [29] Roussas, G.G., Asymptotic normality of random fields of positively or negatively associated processes, J. multivariate anal., 50, 152-173, (1994) · Zbl 0806.60040 [30] 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 [31] Roussas, G.G., 1996. Exponential probability inequalities with some applications. In: Ferguson, T.S., Shapely, L.S., MacQueen, J.B. (Eds.), Statistics, Probability and Game Theory. IMS Lecture Notes-Monograph Series, vol. 30, pp. 303-319, Hayward, CA. [32] Sarkar, S.K.; Chang, C.-K., The simes method for multiple hypothesis testing with positively dependent test statistics, J. amer. statist. assoc., 92, 1601-1608, (1997) · Zbl 0912.62079 [33] Shaked, M., Tong, Y.L., 1992. Stochastic Inequalities. IMS Lecture Notes-Monograph Series, vol. 22, Hayward, CA. [34] Shaked, M., Shanthikumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, Boston. · Zbl 0806.62009 [35] Simon, B., Correlation inequalities and the mass gap in P(ϕ)2: I. domination by the two point function, Commun. math. phys., 31, 127-136, (1973) · Zbl 1125.81313 [36] Szekli, R., 1995. Stochastic Ordering and Dependence in Applied Probability. Springer, New York. · Zbl 0815.60017 [37] Yoshihara, K., 1997. Weakly Dependent Stochastic Sequences and Their Applications. vol. IX, Poisson Approximation and Associated Processes. Sanseido Co., Ltd., Tokyo. · Zbl 0892.60003 [38] 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
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