Mathematics and Computer Science
Volume 4, Issue 2, March 2019, Pages: 57-62
Received: Jul. 26, 2019;
Accepted: Aug. 28, 2019;
Published: Sep. 21, 2019
Views 613 Downloads 120
Wei Li, School of Mathematics & Physics, Hebei University of Engineering, Handan, China
Xiaosheng Wang, School of Mathematics & Physics, Hebei University of Engineering, Handan, China
An uncertain variable is a Borel measurable function whose domain is uncertainty space and range is the set of real numbers. However, for many reasons, like the difficult of collecting data, the value of an uncertain variable is usually not easy to measure accurately. Hence many scholars study the estimation range of the value of an uncertain variable, and usually to estimate the upper or lower bounds of the moment of an uncertain variable is the primary idea. Many inequalities are established to estimate the above bounds, but there are still some problems on the estimation of the moment of uncertain variables. For instance, the even-order moment of an uncertain variable cannot be uniquely calculated at present. So the aim of this paper is to estimate the upper or power bounds of the moment of uncertain variables or the uncertain measure of an event by establishing several new inequalities. Firstly, we extend the Lyapunov inequality on uncertain variable and this inequality gives the upper bound of the even-order moment of an uncertain variable, and as a corollary, the lower bound of the above even-order moment is given. Then the inequality of arithmetic-geometry is proved, which estimates the lower bound of the expected value of an uncertain variable. After that, two equivalent inequalities are given, which can be used to judge the existence of the expected value of a function of an uncertain variable. Finally, as for two independent and identically distributed uncertain variables, the weakly symmetric inequalities are investigated to estimate the upper and lower bounds of the uncertainty distributions of the difference of these uncertain variables which implies the uncertain measures of several events. The above inequalities extend the application range of uncertain variable.
Several Inequalities on Moment of Uncertain Variables, Mathematics and Computer Science.
Vol. 4, No. 2,
2019, pp. 57-62.
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/
) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Z. Lin, Z. Bai, Probability Inequality (in Chinese), Science Press, Beijing, 2006.
P. Chen, B. Peng, S. H. Sung, The von bahrcesseen moment inequality for pairwise independent random variables and applications, Journal of Mathematical Analysis & Applications 419 (2014) 1290–1302.
B. Gavrea, A hermite-hadamard type inequality with applications to the estimation of moments of continuous random variables, Applied Mathematics and Computation 254 (2015) 92–98.
L. Zhang, Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science China Mathematics 59 (2016) 751–768.
P. Gibilisco, F. Hansen, An inequality for expectation of means of positive random variables, Annals of Functional Analysis 8 (2016) 142–151.
C. Pelekis, J. Ramon, Hoeffdings inequality for sums of dependent random variables, Mediterranean Journal of Mathematics 14 (2017) 243.
J. Krebs, A large deviation inequality for β-mixing time series and its applications to the functional kernel regression model, Statistics and Probability Letters 133 (2018) 50–58.
E. Rio, About Doob’s inequality, entropy and Tchebichef, Electronic Communications in Probability 23 (2018) 1–12.
A. Maurer, A Bernstein-type inequality for functions of bounded interaction, Bernoulli 25 (2019) 1451–1471.
B. Liu, Uncertainty Theory, Springer Berlin Heidelberg, Berlin, Heidelberg, 2010, pp. 1–79. URL: https://doi.org/10.1007/978-3-642-13959-8_1.195 doi: 10.1007/978-3-642-13959-8_1.
X. Yang, Moments and tails inequality within the framework of uncertainty theory, International Journal on Information 14 (2010) 2599–2604.
J. Tian, Inequalities and mathematical properties of uncertain variables, Fuzzy Optimization & Decision Making 10 (2011) 357–368.
Q. Chen, Y. Zhu, A class of uncertain variational inequality problems, Journal of Inequalities & Applications (2015) 231.
B. Liu, Uncertainty Theory, 4th edn, Springer-Verlag, Berlin, Heidelberg, 2015.
B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer Publishing Company, Incorporated, Berlin, Heidelberg, 2010.
Y. Sheng, S. Kar, Some results of moments of uncertain variable through inverse uncertainty distribution, Fuzzy Optimization & Decision Making 14 (2015) 57–76.