Elliptic entropy of uncertain random variables with application to portfolio selection
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METHODOLOGIES AND APPLICATION
Elliptic entropy of uncertain random variables with application to portfolio selection Lin Chen1 · Rong Gao2 · Yuxiang Bian3 · Huafei Di4
© Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper investigates an elliptic entropy of uncertain random variables and its application in the area of portfolio selection. We first define the elliptic entropy to characterize the uncertainty of uncertain random variables and give some mathematical properties of the elliptic entropy. Then we derive a computational formula to calculate the elliptic entropy of function of uncertain random variables. Furthermore, we use the elliptic entropy to characterize the risk of investment and establish a mean-entropy portfolio selection model, in which the future security returns are described by uncertain random variables. Based on the chance theory, the equivalent form of the mean–entropy model is derived. To show the performance of the mean–entropy portfolio selection model, several numerical experiments are presented. We also numerically compare the mean–entropy model with the mean–variance model, the equi-weighted portfolio model, and the most diversified portfolio model by using three kinds of diversification indices. Numerical results show that the mean-entropy model outperforms the mean–variance model in selecting diversified portfolios no matter of using which diversification index. Keywords Uncertainty theory · Elliptic entropy · Uncertain random variable · Chance theory · Mean-entropy model · Diversification index
1 Introduction Shannon (1949) first initialized the entropy of random variables in logarithm form. After that, several scholars investigated the entropy in different angles. For example, Kullback and Leibler (1951) presented relative entropy to characterize the degree of difference between two random variables. Jaynes (1957) proposed the principle of maximum entropy and selected the probability distribution with maxCommunicated by V. Loia.
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Rong Gao [email protected] Huafei Di [email protected]
1
College of Management and Economics, Tianjin University, Tianjin 300072, China
2
School of Economics and Management, Hebei University of Technology, Tianjin 300401, China
3
School of Business, East China University of Political Science and Law, Shanghai 201620, China
4
School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
imum entropy from an infinite probability distribution that satisfied the given expected value and variance. Carbone and Stanley (2007) calculated the Shannon entropy of time series by using probability density function of long-range correlation cluster. Ponta and Carbone (2018) used the entropy measurement to implement the time series of prices and fluctuations in financial markets. In the above literature of investigating entropy, a key theoretical assumption is that the indetermination is characterized by random variables (Gao et al. 2017; Rao et al. 2020; Rao and Yan 2020; Xiao et al. 2020). However
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