Variance upper bounds and a probability inequality for discrete α-unimodality

M. Ageel

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 4, page 403-410
  • ISSN: 1233-7234

Abstract

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Variance upper bounds for discrete α-unimodal distributions defined on a finite support are established. These bounds depend on the support and the unimodality index α. They increase as the unimodality index α increases. More information about the underlying distributions yields tighter upper bounds for the variance. A parameter-free Bernstein-type upper bound is derived for the probability that the sum S of n independent and identically distributed discrete α-unimodal random variables exceeds its mean E(S) by a positive value nt. The bound for P{S-nμ ≥ nt} depends on the range of the summands, the sample size n, the unimodality index α and the positive number t.

How to cite

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Ageel, M.. "Variance upper bounds and a probability inequality for discrete α-unimodality." Applicationes Mathematicae 27.4 (2000): 403-410. <http://eudml.org/doc/219283>.

@article{Ageel2000,
abstract = {Variance upper bounds for discrete α-unimodal distributions defined on a finite support are established. These bounds depend on the support and the unimodality index α. They increase as the unimodality index α increases. More information about the underlying distributions yields tighter upper bounds for the variance. A parameter-free Bernstein-type upper bound is derived for the probability that the sum S of n independent and identically distributed discrete α-unimodal random variables exceeds its mean E(S) by a positive value nt. The bound for P\{S-nμ ≥ nt\} depends on the range of the summands, the sample size n, the unimodality index α and the positive number t.},
author = {Ageel, M.},
journal = {Applicationes Mathematicae},
keywords = {probability inequality; variance; upper and lower bounds; discrete unimodality},
language = {eng},
number = {4},
pages = {403-410},
title = {Variance upper bounds and a probability inequality for discrete α-unimodality},
url = {http://eudml.org/doc/219283},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Ageel, M.
TI - Variance upper bounds and a probability inequality for discrete α-unimodality
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 4
SP - 403
EP - 410
AB - Variance upper bounds for discrete α-unimodal distributions defined on a finite support are established. These bounds depend on the support and the unimodality index α. They increase as the unimodality index α increases. More information about the underlying distributions yields tighter upper bounds for the variance. A parameter-free Bernstein-type upper bound is derived for the probability that the sum S of n independent and identically distributed discrete α-unimodal random variables exceeds its mean E(S) by a positive value nt. The bound for P{S-nμ ≥ nt} depends on the range of the summands, the sample size n, the unimodality index α and the positive number t.
LA - eng
KW - probability inequality; variance; upper and lower bounds; discrete unimodality
UR - http://eudml.org/doc/219283
ER -

References

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  3. A. M. Abouammoh and A. F. Mashhour (1994), Variance upper bounds and convolutions of α-unimodal distributions, Statist. Probab. Lett. 21, 281-289. Zbl0809.60023
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  9. S. E. Navard, J. W. Seaman and D. M. Young (1993), A characterization of discrete unimodality with applications to variance upper bounds, Ann. Inst. Statist. Math. 45, 603-614. Zbl0816.62012
  10. R. A. Olshen and L. J. Savage (1970), A generalized unimodality, J. Appl. Probab. 7, 21-34. Zbl0193.45102
  11. J. C. W. Rayner (1975), Variance bounds, Sankhyā Ser. B 37, 135-138. 
  12. J. W. Seaman, P. L. Odell and D. N. Young (1985), Maximum variance for unimodal distributions, Statist. Probab. Lett. 3, 255-260. Zbl0578.62022
  13. J. W. Seaman, D. M. Young and D. W. Turner (1987), On the variance of certain bounded random variables, Math. Sci. 12, 109-116. Zbl0656.62029
  14. F. W. Steutel (1988), Note on discrete α-unimodality, Statist. Neerlandica 42, 137-140. 
  15. D. M. Young, J. W. Seaman, D. W. Turner and V. R. Marco (1988), Probability inequalities for continuous unimodal random variables with finite support, Comm. Statist. Theory Methods 17, 3505-3519. Zbl0696.62035

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