zbMATH — the first resource for mathematics

Large deviations of divergence measures on partitions. (English) Zbl 0996.62052
Summary: We discuss Chemoff-type large deviation results for the total variation, the I-divergence errors, and the \(\chi^2\)-divergence errors on partitions. In contrast to the total variation and the I-divergence, the \(\chi^2\)-divergence has an unconventional large deviation rate. Applications to Bahadur effciencies of goodness-of-fit tests based on these divergence measures for multivariate observations are given.

62G20 Asymptotic properties of nonparametric inference
60F10 Large deviations
62G10 Nonparametric hypothesis testing
Full Text: DOI
[1] Ali, M.S.; Silvey, S.D., A general class of coefficients of divergence of one distribution from another, J. roy. statist. soc. ser B, 28, 131-140, (1966) · Zbl 0203.19902
[2] Bahadur, R.R., Some limit theorems in statistics., (1971), SIAM Philadelphia · Zbl 0257.62015
[3] Barron, A.R., Uniformly powerful goodness of fit tests, Ann. statist., 17, 107-124, (1989) · Zbl 0674.62032
[4] Beirlant, J.; Györfi, L., On the L1-error in histogram density estimation: the multidimensional case, Nonparametric statist., 9, 197-216, (1998) · Zbl 0921.62040
[5] Beirlant, J.; Györfi, L.; Lugosi, G., On the asymptotic normality of the L1- and the L2-errors in histogram density estimation, Canadian J. statist., 22, 309-318, (1995)
[6] Csiszár, I., Information-type measures of divergence of probability distributions and indirect observations, Studia sci. math. hungar., 2, 299-318, (1967) · Zbl 0157.25802
[7] Csiszár, I.; Körner, J., Information theory: coding theorems for memoryless systems., (1981), Academic Press New York · Zbl 0568.94012
[8] Dembo, A., Zeitouni, O., 1992. Large Deviations Techniques and Applications. Jones and Bartlett Publishers, Boston. · Zbl 1177.60035
[9] Devroye, L.; Györfi, L., Nonparametric density estimation: the L1 view., (1985), Wiley New York
[10] Devroye, L.; Györfi, L.; Lugosi, G., A probabilistic theory of pattern recognition., (1996), Springer New York · Zbl 0853.68150
[11] Groeneboom, P.; Shorack, G.R., Large deviations of goodness of fit statistics and linear combinations of order statistics, Ann. probab., 9, 971-987, (1981) · Zbl 0473.60035
[12] Györfi, L.; van der Meulen, E.C., A consistent goodness-of-fit test based on the total variation distance., (), 631-646
[13] Kallenberg, W.C.M., On moderate and large deviations in multinomial distributions, Ann. statist., 13, 1554-1580, (1985) · Zbl 0581.60023
[14] Kallenberg, W.C.M.; Oosterhof, J.; Shriever, B.F., The number of classes in chi-squared goodness-of-fit tests, J. amer. statist. assoc., 80, 959-968, (1985) · Zbl 0582.62037
[15] Kemperman, J.H.B., An optimum rate of transmitting information, Ann. math. statist., 40, 2156-2177, (1969) · Zbl 0287.94021
[16] Kullback, S., A lower bound for discrimination in terms of variation, IEEE trans. inform. theory, 13, 126-127, (1967)
[17] Kullback, S.; Leibler, R.A., On information and sufficiency, Ann. math. statist., 22, 79-86, (1951) · Zbl 0042.38403
[18] Liese, F.; Vajda, I., Convex statistical distances., (1987), Teubner Leipzig · Zbl 0656.62004
[19] Louani, D., Large deviation limit theorems for the kernel density estimator, Scand. J. statist., 25, 243-253, (1998) · Zbl 0904.62060
[20] Louani, D., Large deviations for the L1-distance in kernel density estimation, J. statist. plann. inf., 90, 177-182, (2000) · Zbl 0961.62028
[21] Neyman, J., 1949. Contribution to the theory of the χ2 test. Proceedings of the First Berkeley Symposium on Mathematical Statistics and Probability, Berkeley University Press, Berkeley, CA, pp. 239-273.
[22] Nikitin, Ya., Asymptotic efficiency of nonparametric tests, (1995), Cambridge University Press Cambridge · Zbl 0879.62045
[23] Quine, M.P.; Robinson, J., Efficiencies of chi-square and likelihood ratio goodness-of-fit tests., Ann. statist., 13, 727-742, (1985) · Zbl 0576.62061
[24] Sanov, I.N., 1957. On the probability of large deviations of random variables. Mat. Sb. 42, 11-44 (English translation in Sel. Transl. Math. Statist. Probab. 1 (1961) 213-244).
[25] Toussaint, G.T., Sharper lower bounds for information in term of variation, IEEE trans. inform. theory, IT-21, 99-103, (1975) · Zbl 0293.94012
[26] Tusnády, G., On asymptotically optimal tests, Ann. statist., 5, 385-393, (1977) · Zbl 0361.62034
[27] Vajda, I., Note on discrimination information and variation, IEEE trans. inform. theory, IT-16, 771-773, (1970) · Zbl 0206.21001
[28] Vajda, I., Theory of statistical inference and information., (1989), Kluwer Academic Publishers Dordrecht · Zbl 0678.62035
[29] Watson, G.S., On chi-squared goodness-of-fit tests for continuous distributions, J. roy. statist. soc. ser. B, 20, 44-61, (1958) · Zbl 0086.12701
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.