×

zbMATH — the first resource for mathematics

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:
60E15 Inequalities; stochastic orderings
PDF BibTeX XML Cite
Full Text: DOI
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
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.