Kybernetika 52 no. 6, 943-966, 2016

Bounds on tail probabilities for negative binomial distributions

This article was granted Editor's award of the year 2016Editor's award 2016

Peter HarremoësDOI: 10.14736/kyb-2016-6-0943


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.


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


60E15, 62E17, 60F10


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