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doi: 10.3934/jimo.2020163
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## The skewness for uncertain random variable and application to portfolio selection problem

 1 School of Applied Mathematics, Nanjing University of Finance and Economics, Nanjing 210023, China 2 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Bo Li

Received  June 2020 Revised  August 2020 Early access November 2020

Uncertainty and randomness are two basic types of indeterminacy, where uncertain variable is used to represent quantities with human uncertainty and random variable is applied for modeling quantities with objective randomness. In many real systems, uncertainty and randomness often exist simultaneously. Then uncertain random variable and chance measure can be used to handle such cases. We know that the skewness is a measure of distributional asymmetry. However, the concept of skewness for uncertain random variable has not been clearly defined. In this paper, we first propose a concept of skewness for uncertain random variable and then present a formula for calculating the skewness via chance distribution. Applying the presented formula, the skewnesses of three special uncertain random variables are derived. Finally, a portfolio selection problem is carried out for showing the efficiency and applicability of skewness and presented formula.

Citation: Bo Li, Yadong Shu. The skewness for uncertain random variable and application to portfolio selection problem. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020163
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##### References:
The computational results for different $p$ and $q$
 $(p, q)$ $(x_{1}^{*}, x_{2}^{*})$ Expected value Variance Skewness $(0.04, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.04, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.04, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$
 $(p, q)$ $(x_{1}^{*}, x_{2}^{*})$ Expected value Variance Skewness $(0.04, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.04, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.04, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.02, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.2)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.5)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$ $(0.01, 0.8)$ $(0, 1)$ $0.04$ 0.0075 $1.125\times10^{-4}$
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