Bounds on tail probabilities for negative binomial distributions

Peter Harremoës

Kybernetika (2016)

  • Volume: 52, Issue: 6, page 943-966
  • ISSN: 0023-5954

Abstract

top
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.

How to cite

top

Harremoës, Peter. "Bounds on tail probabilities for negative binomial distributions." Kybernetika 52.6 (2016): 943-966. <http://eudml.org/doc/287922>.

@article{Harremoës2016,
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.},
author = {Harremoës, Peter},
journal = {Kybernetika},
keywords = {tail probability; exponential family; signed log-likelihood; variance function; inequalities},
language = {eng},
number = {6},
pages = {943-966},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Bounds on tail probabilities for negative binomial distributions},
url = {http://eudml.org/doc/287922},
volume = {52},
year = {2016},
}

TY - JOUR
AU - Harremoës, Peter
TI - Bounds on tail probabilities for negative binomial distributions
JO - Kybernetika
PY - 2016
PB - Institute of Information Theory and Automation AS CR
VL - 52
IS - 6
SP - 943
EP - 966
AB - 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.
LA - eng
KW - tail probability; exponential family; signed log-likelihood; variance function; inequalities
UR - http://eudml.org/doc/287922
ER -

References

top
  1. Alfers, D., Dinges, H., 10.1007/bf00533744, Z. Wahrscheinlichkeitstheory verw. Geb. 65 (1984), 3, 399-420. Zbl0506.62011MR0731229DOI10.1007/bf00533744
  2. Bahadur, R. R., 10.1214/aoms/1177705986, Ann. Math. Statist. 31 (1960), 43-54. Zbl0092.35203MR0120675DOI10.1214/aoms/1177705986
  3. Bahadur, R. R., Rao, R. R., 10.1214/aoms/1177705674, Ann. Math. Statist. 31 (1960), 4, 1015-1027. MR0117775DOI10.1214/aoms/1177705674
  4. Barndorff-Nielsen, O. E., A note on the standardized signed log likelihood ratio., Scand. J. Statist. 17 (1990), 2, 157-160. Zbl0716.62021MR1085928
  5. Györfi, L., Harremoës, P., Tusnády, G., Gaussian approximation of large deviation probabilities., http://www.harremoes.dk/Peter/ITWGauss.pdf, 2012. 
  6. Harremoës, P., 10.1109/isit.2014.6875279, In: Proc. ISIT 2014, IEEE 2014, pp. 2474-2478. DOI10.1109/isit.2014.6875279
  7. Harremoës, P., Tusnády, G., 10.1109/isit.2012.6284247, In: International Symposium on Information Theory (ISIT 2012) (Cambridge, Massachusetts), IEEE 2012, pp. 538-543. DOI10.1109/isit.2012.6284247
  8. Letac, G., Mora, M., 10.1214/aos/1176347491, Ann. Stat. 18 (1990), 1, 1-37. Zbl0714.62010MR1041384DOI10.1214/aos/1176347491
  9. Morris, C., 10.1214/aos/1176345690, Ann. Statist. 10 (1982), 65-80. Zbl0521.62014MR0642719DOI10.1214/aos/1176345690
  10. Reiczigel, J., Rejtő, L., Tusnády, G., A sharpning of Tusnády's inequality., arXiv: 1110.3627v2, 2011. 
  11. Zubkov, A. M., Serov, A. A., 10.1137/s0040585x97986138, Theory Probab. Appl. 57 (2013), 3, 539-544. Zbl1280.60016MR3196787DOI10.1137/s0040585x97986138

NotesEmbed ?

top

You must be logged in to post comments.