A New Characterization of Exponential Distribution through Minimum Chi-Squared Divergence Principle
Abstract
In this paper, a new characterizing theorems of exponential distribution based on minimum chi-Squared divergence principle are presented. We study minimum chi-squared divergence probability distributions by this principle given a prior exponential distribution and the available information on moments. Some illustrative examples are included for special values of the parameters. We tabulated the results and the corresponding characteristics of the distribution are graphically, compared.
How to cite this article: Kamel ARR, Alqarni AAAA. A New Characterization of Exponential Distribution through Minimum Chi-Squared Divergence Principle. J Adv Res Appl Math Stat 2020; 5(1&2): 14-26.
DOI: https://doi.org/10.24321/2455.7021.202002
References
1. Abonazel M, Rabie A. The impact of using robust estimations in regression models: An application on the Egyptian economy. Journal of Advanced Research in Applied Mathematics and Statistics 2019; 4(2): 8-16.
2. Abonazel M, Gad A. Robust partial residuals estimation in semiparametric partially linear model. Communications in Statistics-Simulation and Computation 2018: 1-14.
3. Ahsanullah M, Hamedani G. Exponential Distribution: Theory and Methods, NOVA Science 2010, New York.
4. Bairamov I, Özkal T. On characterization of distributions through the properties of conditional expectations of order statistics. Communications in Statistics Theory and Methods 2007; 36(7): 1319-1326.
5. Campbell L. Equivalence of Gauss’s principle and minimum discrimination information estimation of probabilities. The Annals of Mathematical Statistics 1970; 41(3): 1011-1015.
6. Yanev G, Chakrabort S. Characterizations of exponential distribution based on sample of size three. Pliska Studia Mathematica Bulgarica 2013; 22(1): 237-244.
7. Gertsbak I, Kagan A. Characterization of the weibull distribution by properties of the Fisher information under type-I censoring. Statistics and probability letters 1999; 42(1): 99-105.
8. Gupta A, Nguyen T, Zeng W. Characterization of multivariate distributions through a functional equation of their characteristic functions. Journal of statistical planning and inference 1997; 63(2): 187-201.
9. Hamedani G, Ahsanullah M. Characterizations of weibull geometric distribution. Journal of Statistical Theory and Applications 2011; 10(4): 581-590.
10. Januškevičius R. Stability characterizations of some probability distributions. LAP LAMBERT Academic Publishing 2014.
11. Jaynes E. Information theory and statistical mechanics. Physical review 1957; 106(4): 620.
12. Kapur J. Maximum-entropy probability distribution for a continuous random variate over a finite interval. J. Math. Phys. Sci 1982; 16(1): 97-109.
13. Kapur J. Twenty-five years of maximum-entropy principle. Journal of Mathematical and Physical Sciences 1983; 17: 103-156.
14. Kawamura T, Iwase K. Characterizations of the distributions of power inverse Gaussian and others based on the entropy maximization principle. Journal of the Japan Statistical Society 2003; 33(1): 95-104.
15. Kesavan H, Kapur J. The generalized maximum entropy principle. IEEE Transactions on systems, Man, and Cybernetics 1989; 19(5): 1042-1052.
16. Khan A, Faizan M, Haque Z. Characterization of probability distributions through order statistics. In ProbStat Forum 2009; 2: 132-136.
17. Kullback S. Information theory and statistics. Courier corporation. New York, John Wiley, 1997.
18. Kumar P. Characterizations of beta probability distributions based on the minimum χ2–Divergence principle 2005.
19. Kumar P. Probability distributions conditioned by the available information: Gamma distribution and moments. Computers and Mathematics with
Applications 2006; 52(3&4): 289-304.
20. Kumar P, Taneja I. Chi square divergence and minimization problem. J. Comb. Inf. Syst. Sci 2004; 28: 181-207.
21. Ahsanullah M, Rahman A. Characterization of the exponential distribution. J Appl Prob 1972; 9(2): 457-461.
22. Ord J. Characterization problems in mathematical statistics. Journal of the Royal Statistical Society: Series A (General) 1975; 138(4): 576-577.
23. Shore J, Johnson R. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Transactions on information theory 1980; 26(1): 26-37.
24. Tavangar M, Asadi M. Some new characterization results on exponential and related distributions 2010; 257-272.
25. Yanev G. Chakraborty, S. A characterization of exponential distribution and the SukhatmeRnyi decomposition of exponential maxima, Statistics and Probability Letters 2016; 110: 94 - 102.
26. Zolotarev V. Metric distances in spaces of random variables and their distributions. Mathematics of the USSR-Sbornik 1976; 30(3): 373.
2. Abonazel M, Gad A. Robust partial residuals estimation in semiparametric partially linear model. Communications in Statistics-Simulation and Computation 2018: 1-14.
3. Ahsanullah M, Hamedani G. Exponential Distribution: Theory and Methods, NOVA Science 2010, New York.
4. Bairamov I, Özkal T. On characterization of distributions through the properties of conditional expectations of order statistics. Communications in Statistics Theory and Methods 2007; 36(7): 1319-1326.
5. Campbell L. Equivalence of Gauss’s principle and minimum discrimination information estimation of probabilities. The Annals of Mathematical Statistics 1970; 41(3): 1011-1015.
6. Yanev G, Chakrabort S. Characterizations of exponential distribution based on sample of size three. Pliska Studia Mathematica Bulgarica 2013; 22(1): 237-244.
7. Gertsbak I, Kagan A. Characterization of the weibull distribution by properties of the Fisher information under type-I censoring. Statistics and probability letters 1999; 42(1): 99-105.
8. Gupta A, Nguyen T, Zeng W. Characterization of multivariate distributions through a functional equation of their characteristic functions. Journal of statistical planning and inference 1997; 63(2): 187-201.
9. Hamedani G, Ahsanullah M. Characterizations of weibull geometric distribution. Journal of Statistical Theory and Applications 2011; 10(4): 581-590.
10. Januškevičius R. Stability characterizations of some probability distributions. LAP LAMBERT Academic Publishing 2014.
11. Jaynes E. Information theory and statistical mechanics. Physical review 1957; 106(4): 620.
12. Kapur J. Maximum-entropy probability distribution for a continuous random variate over a finite interval. J. Math. Phys. Sci 1982; 16(1): 97-109.
13. Kapur J. Twenty-five years of maximum-entropy principle. Journal of Mathematical and Physical Sciences 1983; 17: 103-156.
14. Kawamura T, Iwase K. Characterizations of the distributions of power inverse Gaussian and others based on the entropy maximization principle. Journal of the Japan Statistical Society 2003; 33(1): 95-104.
15. Kesavan H, Kapur J. The generalized maximum entropy principle. IEEE Transactions on systems, Man, and Cybernetics 1989; 19(5): 1042-1052.
16. Khan A, Faizan M, Haque Z. Characterization of probability distributions through order statistics. In ProbStat Forum 2009; 2: 132-136.
17. Kullback S. Information theory and statistics. Courier corporation. New York, John Wiley, 1997.
18. Kumar P. Characterizations of beta probability distributions based on the minimum χ2–Divergence principle 2005.
19. Kumar P. Probability distributions conditioned by the available information: Gamma distribution and moments. Computers and Mathematics with
Applications 2006; 52(3&4): 289-304.
20. Kumar P, Taneja I. Chi square divergence and minimization problem. J. Comb. Inf. Syst. Sci 2004; 28: 181-207.
21. Ahsanullah M, Rahman A. Characterization of the exponential distribution. J Appl Prob 1972; 9(2): 457-461.
22. Ord J. Characterization problems in mathematical statistics. Journal of the Royal Statistical Society: Series A (General) 1975; 138(4): 576-577.
23. Shore J, Johnson R. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Transactions on information theory 1980; 26(1): 26-37.
24. Tavangar M, Asadi M. Some new characterization results on exponential and related distributions 2010; 257-272.
25. Yanev G. Chakraborty, S. A characterization of exponential distribution and the SukhatmeRnyi decomposition of exponential maxima, Statistics and Probability Letters 2016; 110: 94 - 102.
26. Zolotarev V. Metric distances in spaces of random variables and their distributions. Mathematics of the USSR-Sbornik 1976; 30(3): 373.
Published
2020-12-28
How to Cite
KAMEL, Amr Ragab Rabie; ALQARNI, Abd AlAziz Abd Allah.
A New Characterization of Exponential Distribution through Minimum Chi-Squared Divergence Principle.
Journal of Advanced Research in Applied Mathematics and Statistics, [S.l.], v. 5, n. 1&2, p. 14-26, dec. 2020.
ISSN 2455-7021.
Available at: <http://thejournalshouse.com/index.php/Journal-Maths-Stats/article/view/13>. Date accessed: 30 jan. 2025.
Section
Research Article
