| Title:
|
On an inequality and the related characterization of the gamma distribution (English) |
| Author:
|
Koicheva, Maia |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
38 |
| Issue:
|
1 |
| Year:
|
1993 |
| Pages:
|
11-18 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we derive conditions upon the nonnegative random variable \xi under which the inequality $Dg(\xi)\leq cE\left[g'\left(\xi\right)\right]^2\xi$ holds for a fixed nonnegative constant $c$ and for any absolutely continuous function $g$. Taking into account the characterization of a Gamma distribution we consider the functional $U_\xi = \sup_g \frac{Dg\left(\xi\right)}{E\left[g'\left(\xi\right)\right]^2\xi}$ and establishing some of its properties we show that $U_\xi \geq 1$ and that $U_\xi =1$ iff the random variable $\xi$ has a Gamma distribution. (English) |
| Keyword:
|
characterizations |
| Keyword:
|
Gamma distribution |
| MSC:
|
60E15 |
| MSC:
|
60E99 |
| MSC:
|
62E10 |
| idZBL:
|
Zbl 0776.60021 |
| idMR:
|
MR1202076 |
| DOI:
|
10.21136/AM.1993.104530 |
| . |
| Date available:
|
2008-05-20T18:44:46Z |
| Last updated:
|
2020-07-28 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/104530 |
| . |
| Reference:
|
[1] H. Chernoff: A note on an inequality involving the normal distribution.Ann. Probab. 9 (3) (1981), 533-535. Zbl 0457.60014, MR 0614640, 10.1214/aop/1176994428 |
| Reference:
|
[2] A. A. Borovkov S. A. Utev: On an inequality and a related characterization of the normal distribution.Theory of Probab. and its Appl. 28 (2) (1983), 219-228. MR 0700206 |
| Reference:
|
[3] T. Cacoullos: On upper and lower bounds for the variance of a function of a random variable.Ann. Probab. 10(1982), 799-809. Zbl 0492.60021, MR 0659549, 10.1214/aop/1176993788 |
| Reference:
|
[4] T. Cacoullos V. Papathanasiou: On upper bounds for the variance of functions of random variables.Statistics and Probability Letters 3 (1985), 175-184. MR 0801687, 10.1016/0167-7152(85)90014-8 |
| Reference:
|
[5] L. Chen: An inequality for the multivariate normal distribution.J. Multivariate Anal. 12 (1982), 306-315. Zbl 0483.60011, MR 0661566, 10.1016/0047-259X(82)90022-7 |
| Reference:
|
[6] B. L. S. Prakasa Rao M. Sreehari: Another characterization of multivariate normal distribution.Statistics and Probability Letters 4 (1986), 209-210. MR 0848719, 10.1016/0167-7152(86)90068-4 |
| Reference:
|
[7] B. L. S. Prakasa Rao M. Sreehari: On a characterization of Poisson distribution through an inequality of Chernoff type.Aus. J. Statist. 29 (1987), 38-41. MR 0899374, 10.1111/j.1467-842X.1987.tb00718.x |
| Reference:
|
[8] T. Cacoullos V. Papathanasiou: Characterizations of distributions by variance bounds.Statistics and Probability Letters 7 (5) (1989), 351-356. MR 1001133, 10.1016/0167-7152(89)90050-3 |
| . |
