Geodesic
Bounds on tail probabilities for negative binomial distributions
Kybernetika, Tome 52 (2016) no. 6, pp. 943-966 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.14736/kyb-2016-6-0943
Classification : 60E15, 60F10, 62E17
Keywords: tail probability; exponential family; signed log-likelihood; variance function; inequalities 
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     author = {Harremo\"es, Peter},
     title = {Bounds on tail probabilities for negative binomial distributions},
     journal = {Kybernetika},
     pages = {943--966},
     year = {2016},
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     zbl = {06707382},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.14736/kyb-2016-6-0943/}
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Harremoës, Peter. Bounds on tail probabilities for negative binomial distributions. Kybernetika, Tome 52 (2016) no. 6, pp. 943-966. doi: 10.14736/kyb-2016-6-0943

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