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

Applicationes Mathematicae (2000)

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

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topAgeel, 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

top- A. M. Abouammoh and A. F. Mashhour (1983), On characterization of discrete unimodality: a survey, in: Developments in Statistics and its Applications, KSU Libraries, 327-342.
- A. M. Abouammoh, A. M. Ali and A. F. Mashhour (1994), On characterizations and variance bounds of discrete α-unimodality, Statist. Papers 35, 151-161. Zbl0807.62008
- 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
- M. H. Alamatsaz (1985), A note on an article by Artikis, Acta Math. Hungar. 45, 159-162. Zbl0601.60011
- S. W. Dharmadhikari and K. Jogdeo (1986), Some results on generalized unimodality and an application to Chebyshev's inequality, in: Reliability and Quality Control, Elsevier, 127-132.
- S. W. Dharmadhikari and K. Joag-Dev (1989), Upper bounds for the variances of certain random variables, Comm. Statist. Theory Methods 18, 3235-3247. Zbl0696.62036
- M. I. Jacobson (1969), The maximum variance of restricted unimodal distributions, Ann. Math. Statist. 40, 1746-1752. Zbl0198.23802
- J. Muilwijk (1966), Note on a theorem of M. N. Murthy and V. K. Sethi, Sankhyā Ser. B 28, 183.
- 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
- R. A. Olshen and L. J. Savage (1970), A generalized unimodality, J. Appl. Probab. 7, 21-34. Zbl0193.45102
- J. C. W. Rayner (1975), Variance bounds, Sankhyā Ser. B 37, 135-138.
- J. W. Seaman, P. L. Odell and D. N. Young (1985), Maximum variance for unimodal distributions, Statist. Probab. Lett. 3, 255-260. Zbl0578.62022
- 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
- F. W. Steutel (1988), Note on discrete α-unimodality, Statist. Neerlandica 42, 137-140.
- 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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