Peter Harremoës

# Abstract:

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.

# Keywords:

exponential family, variance function, tail probability, signed log-likelihood, inequalities

# Classification:

60E15, 62E17, 60F10

# References:

1. D. Alfers and H. Dinges: A normal approximation for beta and gamma tail probabilities. Z. Wahrscheinlichkeitstheory verw. Geb. 65 (1984), 3, 399-420.   DOI:10.1007/bf00533744
2. R. R. Bahadur: Some approximations to the binomial distribution function. Ann. Math. Statist. 31 (1960), 43-54.   DOI:10.1214/aoms/1177705986
3. R. R. Bahadur and R. R. Rao: On deviation of the sample mean. Ann. Math. Statist. 31 (1960), 4, 1015-1027.   DOI:10.1214/aoms/1177705674
4. O. E. Barndorff-Nielsen: A note on the standardized signed log likelihood ratio. Scand. J. Statist. 17 (1990), 2, 157-160.   CrossRef
5. L. Györfi, P. Harremoës and G. Tusnády: Gaussian approximation of large deviation probabilities. http://www.harremoes.dk/Peter/ITWGauss.pdf, 2012.   CrossRef
6. P. Harremoës: Mutual information of contingency tables and related inequalities. In: Proc. ISIT 2014, IEEE 2014, pp. 2474-2478.   DOI:10.1109/isit.2014.6875279
7. P. Harremoës and G. Tusnády: 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. G. Letac and M. Mora: Natural real exponential families with cubic variance functions. Ann. Stat. 18 (1990), 1, 1-37.   DOI:10.1214/aos/1176347491
9. C. Morris: Natural exponential families with quadratic variance functions. Ann. Statist. 10 (1982), 65-80.   DOI:10.1214/aos/1176345690
10. J. Reiczigel, L. Rejtő and G. Tusnády: A sharpning of {T}usn{á}dy's inequality. arXiv: 1110.3627v2, 2011.   CrossRef
11. A. M. Zubkov and A. A. Serov: 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