# Article

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Keywords:
tail probability; exponential family; signed log-likelihood; variance function; inequalities
Summary:
In this paper we derive various bounds on tail probabilities of distributions for which the generated exponential family has a linear or quadratic variance function. The main result is an inequality relating the signed log-likelihood of a negative binomial distribution with the signed log-likelihood of a Gamma distribution. This bound leads to a new bound on the signed log-likelihood of a binomial distribution compared with a Poisson distribution that can be used to prove an intersection property of the signed log-likelihood of a binomial distribution compared with a standard Gaussian distribution. All the derived inequalities are related and they are all of a qualitative nature that can be formulated via stochastic domination or a certain intersection property.
References:
[1] Alfers, D., Dinges, H.: A normal approximation for beta and gamma tail probabilities. Z. Wahrscheinlichkeitstheory verw. Geb. 65 (1984), 3, 399-420. DOI 10.1007/bf00533744 | MR 0731229 | Zbl 0506.62011
[2] Bahadur, R. R.: Some approximations to the binomial distribution function. Ann. Math. Statist. 31 (1960), 43-54. DOI 10.1214/aoms/1177705986 | MR 0120675 | Zbl 0092.35203
[3] Bahadur, R. R., Rao, R. R.: On deviation of the sample mean. Ann. Math. Statist. 31 (1960), 4, 1015-1027. DOI 10.1214/aoms/1177705674 | MR 0117775
[4] Barndorff-Nielsen, O. E.: A note on the standardized signed log likelihood ratio. Scand. J. Statist. 17 (1990), 2, 157-160. MR 1085928 | Zbl 0716.62021
[5] Györfi, L., Harremoës, P., Tusnády, G.: Gaussian approximation of large deviation probabilities. http://www.harremoes.dk/Peter/ITWGauss.pdf, 2012.
[6] Harremoës, P.: Mutual information of contingency tables and related inequalities. In: Proc. ISIT 2014, IEEE 2014, pp. 2474-2478. DOI 10.1109/isit.2014.6875279
[7] Harremoës, P., Tusnády, G.: Information divergence is more $\chi^2$-distributed than the $\chi^2$-statistic. In: International Symposium on Information Theory (ISIT 2012) (Cambridge, Massachusetts), IEEE 2012, pp. 538-543. DOI 10.1109/isit.2012.6284247
[8] Letac, G., Mora, M.: Natural real exponential families with cubic variance functions. Ann. Stat. 18 (1990), 1, 1-37. DOI 10.1214/aos/1176347491 | MR 1041384 | Zbl 0714.62010
[9] Morris, C.: Natural exponential families with quadratic variance functions. Ann. Statist. 10 (1982), 65-80. DOI 10.1214/aos/1176345690 | MR 0642719 | Zbl 0521.62014
[10] Reiczigel, J., Rejtő, L., Tusnády, G.: A sharpning of Tusnády's inequality. arXiv: 1110.3627v2, 2011.
[11] Zubkov, A. M., Serov, A. A.: A complete proof of universal inequalities for the distribution function of the binomial law. Theory Probab. Appl. 57 (2013), 3, 539-544. DOI 10.1137/s0040585x97986138 | MR 3196787 | Zbl 1280.60016

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