Several Inequalities on Moment of Uncertain Variables

Wei Li; Xiaosheng Wang

doi:doi:10.11648/j.mcs.20190402.12

| Peer-Reviewed

Several Inequalities on Moment of Uncertain Variables

Wei Li, Xiaosheng Wang

Published in Mathematics and Computer Science (Volume 4, Issue 2)

Received: 26 July 2019 Accepted: 28 August 2019 Published: 21 September 2019

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

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.

Published in	Mathematics and Computer Science (Volume 4, Issue 2)
DOI	10.11648/j.mcs.20190402.12
Page(s)	57-62
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Inequality, Uncertainty Theory, Upper or Lower Bound

References

[1]	Z. Lin, Z. Bai, Probability Inequality (in Chinese), Science Press, Beijing, 2006.
[2]	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.
[3]	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.
[4]	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.
[5]	P. Gibilisco, F. Hansen, An inequality for expectation of means of positive random variables, Annals of Functional Analysis 8 (2016) 142–151.
[6]	C. Pelekis, J. Ramon, Hoeffdings inequality for sums of dependent random variables, Mediterranean Journal of Mathematics 14 (2017) 243.
[7]	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.
[8]	E. Rio, About Doob’s inequality, entropy and Tchebichef, Electronic Communications in Probability 23 (2018) 1–12.
[9]	A. Maurer, A Bernstein-type inequality for functions of bounded interaction, Bernoulli 25 (2019) 1451–1471.
[10]	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.
[11]	X. Yang, Moments and tails inequality within the framework of uncertainty theory, International Journal on Information 14 (2010) 2599–2604.
[12]	J. Tian, Inequalities and mathematical properties of uncertain variables, Fuzzy Optimization & Decision Making 10 (2011) 357–368.
[13]	Q. Chen, Y. Zhu, A class of uncertain variational inequality problems, Journal of Inequalities & Applications (2015) 231.
[14]	B. Liu, Uncertainty Theory, 4th edn, Springer-Verlag, Berlin, Heidelberg, 2015.
[15]	B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer Publishing Company, Incorporated, Berlin, Heidelberg, 2010.
[16]	Y. Sheng, S. Kar, Some results of moments of uncertain variable through inverse uncertainty distribution, Fuzzy Optimization & Decision Making 14 (2015) 57–76.

Cite This Article

Plain Text BibTeX RIS

APA Style

Wei Li, Xiaosheng Wang. (2019). Several Inequalities on Moment of Uncertain Variables. Mathematics and Computer Science, 4(2), 57-62. https://doi.org/10.11648/j.mcs.20190402.12

Copy | Download

ACS Style

Wei Li; Xiaosheng Wang. Several Inequalities on Moment of Uncertain Variables. Math. Comput. Sci. 2019, 4(2), 57-62. doi: 10.11648/j.mcs.20190402.12

Copy | Download

AMA Style

Wei Li, Xiaosheng Wang. Several Inequalities on Moment of Uncertain Variables. Math Comput Sci. 2019;4(2):57-62. doi: 10.11648/j.mcs.20190402.12

Copy | Download

@article{10.11648/j.mcs.20190402.12,
  author = {Wei Li and Xiaosheng Wang},
  title = {Several Inequalities on Moment of Uncertain Variables},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {2},
  pages = {57-62},
  doi = {10.11648/j.mcs.20190402.12},
  url = {https://doi.org/10.11648/j.mcs.20190402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190402.12},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Several Inequalities on Moment of Uncertain Variables
AU  - Wei Li
AU  - Xiaosheng Wang
Y1  - 2019/09/21
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20190402.12
DO  - 10.11648/j.mcs.20190402.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 57
EP  - 62
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20190402.12
AB  - 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.
VL  - 4
IS  - 2
ER  -

Copy | Download

Author Information

Wei Li

School of Mathematics & Physics, Hebei University of Engineering, Handan, China
Xiaosheng Wang

School of Mathematics & Physics, Hebei University of Engineering, Handan, China

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Wei Li, Xiaosheng Wang. (2019). Several Inequalities on Moment of Uncertain Variables. Mathematics and Computer Science, 4(2), 57-62. https://doi.org/10.11648/j.mcs.20190402.12

Copy | Download

ACS Style

Wei Li; Xiaosheng Wang. Several Inequalities on Moment of Uncertain Variables. Math. Comput. Sci. 2019, 4(2), 57-62. doi: 10.11648/j.mcs.20190402.12

Copy | Download

AMA Style

Wei Li, Xiaosheng Wang. Several Inequalities on Moment of Uncertain Variables. Math Comput Sci. 2019;4(2):57-62. doi: 10.11648/j.mcs.20190402.12

Copy | Download

@article{10.11648/j.mcs.20190402.12,
  author = {Wei Li and Xiaosheng Wang},
  title = {Several Inequalities on Moment of Uncertain Variables},
  journal = {Mathematics and Computer Science},
  volume = {4},
  number = {2},
  pages = {57-62},
  doi = {10.11648/j.mcs.20190402.12},
  url = {https://doi.org/10.11648/j.mcs.20190402.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20190402.12},
  abstract = {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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Several Inequalities on Moment of Uncertain Variables
AU  - Wei Li
AU  - Xiaosheng Wang
Y1  - 2019/09/21
PY  - 2019
N1  - https://doi.org/10.11648/j.mcs.20190402.12
DO  - 10.11648/j.mcs.20190402.12
T2  - Mathematics and Computer Science
JF  - Mathematics and Computer Science
JO  - Mathematics and Computer Science
SP  - 57
EP  - 62
PB  - Science Publishing Group
SN  - 2575-6028
UR  - https://doi.org/10.11648/j.mcs.20190402.12
AB  - 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.
VL  - 4
IS  - 2
ER  -

Copy | Download