Geodesic
Bounds on tail probabilities for negative binomial distributions
Kybernetika, Tome 52 (2016) no. 6, pp. 943-966
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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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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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