Partial Gini Coefficient for Uncertain Random Variables with Application to Portfolio Selection

Abstract

The partial Gini coefficient measures the strength of dispersion for uncertain random variables, while controlling for the effects of all random variables. Similarly to variance, the partial Gini coefficient plays an important role in uncertain random portfolio selection problems, as a risk measure to find the optimal proportions for securities. We first define the partial Gini coefficient as a risk measure in uncertain random environments. Then, we obtain a computational formula for computing the partial Gini coefficient of uncertain random variables. Moreover, we apply the partial Gini coefficient to characterize risk of investment and investigate a mean-partial Gini model with uncertain random returns. To display the performance of the mean-partial Gini portfolio selection model, some computational examples are provided. To compare the mean-partial Gini model with the traditional mean-variance model using performance ratio and diversification indices, we apply Wilcoxon non-parametric tests for related samples.

1. Introduction

The Gini coefficient was introduced by Corrado Gini [1] as the expected distance or difference between two independent random variables, to measure income dispersion. A higher Gini coefficient implies greater inequality. Yitzhaki [2] established several representative formulas for the Gini coefficient. After that, Lerman and Yitzhaki [3] proposed the concept of an extended Gini coefficient for random variables and provided a formula for estimating the extended Gini coefficient using the concept of covariance. Furthermore, Chotikapanich and Griffiths [4] presented another estimator for extended Gini coefficient using a Lorenz curve. As an application of the Gini coefficient in finance, Yitzhaki [2] presented a mean-Gini portfolio selection model with random returns by considering the Gini coefficient as a risk measure. Shalit and Yitzhaki [5] presented a modification of the mean-Gini portfolio selection model with random returns. By considering other risk measures, several authors have devoted studies to the case of portfolio selection in random environments. The first portfolio selection problem was proposed by Markowitz [6] using a mean-variance model with random returns. In addition, Konno and Yamazaki [7], Agouram and Lakhnati [8] and Favre and Galeano [9] obtained optimal solutions for random portfolio selection problems using absolute deviation, value at risk, and modified value at risk, respectively.

Though randomness is a basic type of indeterminate phenomena, the indeterminacy in real life is varied. Sometimes, indeterminacy does not behave randomly. Huang [10] discussed an example that can be fitted using uncertain variables as follows: The event chance of a security cost falling within [105, 115] is $30\%$, and the chance of the security price being within [115, 125] is $20\%$. Then, what is the occurrence chance of the security price being in the interval [105, 125]? A general view appears that some individuals accept the chance ought not to be lower than $30\%$ but not more than $50\%$. Moreover, sometimes we have enough historical data but cannot make a decision based on this data. For example, the COVID-19 pandemic and Ukraine war have affected the returns of stock markets. In these situations, the returns are not random. Uncertainty theory, which was founded by Liu [11], established a new tool to handle this type of indeterminacy. In the past decades, researchers have proposed various risk measures for uncertain environments, such as variance (Liu [11], Sheng and Kar [12]), entropy (Liu [11], Dai and Chen [13]),quadratic entropy (Dai [14]), sine entropy (Yao et al. [15]), and cross entropy (Chen et al. [16]). In order to characterize the association between two uncertain variables, Zhao et al. [17] proposed the concept of covariance for two uncertain variables. By applying the concept of covariance, Gao et al. [18] defined the concept of Gini coefficient for an uncertain variable. Furthermore, they derived a formula for computing the Gini coefficient of uncertain variables through inverse uncertainty distribution. Since risk measures play important roles in the decision-making of portfolio optimization, we recall several different portfolio optimization problems. By exploiting a mean-variance model, Qin et al. [19] obtained the optimal proportions for portfolio selection problems, with uncertain returns. Furthermore, a mean-value at risk model was established by Kar et al. [20] through assigning a value at risk and sharp ratio to the risk measure of uncertain returns. As a down risk measure, Qin et al. [21] optimized uncertain portfolio selection problems using semi-absolute deviation. By considering logarithm entropy as a risk measure, Zhou [22] established mean-entropy portfolio allocation models with uncertain returns. Recently, Gao et al. [18] proposed a mean-Gini portfolio selection problem with uncertain returns and compared it with the mean-variance model.

In numerous situations, uncertainty and randomness appear at the same time in a complex framework. For example, there are some existing securities with sufficient frequencies and a few recently recorded ones with inadequate information within the portfolio.In order to model systems, Liu [23] proposed the concept of uncertain random variables using chance theory. As a risk measure, variance of uncertain random variables was presented by Liu [23]. After that, some important contributions were devoted to the application of uncertain random variables, for instance [24,25,26,27,28,29,30]. In the past decades, researchers have proposed various risk measures for uncertain random environments such as variance (Liu [23]), entropy (Sheng et al. [31]), partial entropy (Ahmadzade et al. [32]; Yang and Zhu [33]), and covariance (Ahmadzade and Gao [34]).

Although the above researchers studied several different risk measures for uncertain random variables, some research gaps exist. For instance, the concept of the Gini coefficient has not been studied in chance theory. In this paper, we introduce the concept of the partial Gini coefficient for uncertain random variables and investigate its properties using inverse uncertainty distributions. In the application of risk measures to uncertain random environments, several authors have applied different risk measures to portfolio selection problems [35,36]. The traditional mean-variance portfolio allocation model was proposed by Qin et al. [37]. By introducing the concept of the covariance of two uncertain random variables, Ahmadzade and Gao [34] investigated mean-variance portfolio selection models with uncertain random variables as a quadratic programming model. By defining a risk curve for uncertain random variables, Mehralizade et al. [38] presented mean-risk portfolio selection models with uncertain random returns. Qin et al. [39] established mean-value at risk portfolio selection models, with respect to chance theory. As a different approach, Liu et al. [40] solved mean-value at risk portfolio selection problems using uncertain random returns and Monte-Carlo simulation. Chennaf and Ben Amor [41] proposed the concept of entropic value at risk for uncertain random variables and applied it to portfolio selection problems. By considering entropy as a risk measure, Ahmadzade et al. [42] optimized portfolio allocation problems with mean-triangular entropy models. Furthermore, He et al. [43] investigated the concept of Tsallis entropy for uncertain random variables as a flexible form of logarithm entropy and used it in mean-entropy models. In addition, by minimizing the divergence measure, Ahmadzade et al. [44] derived the best proportions for mean-divergence portfolio selection models. Similarly, Gao et al. [45] found optimal proportions for mean-similarity portfolio selection models through maximizing similarity measures of uncertain random returns. Cheng et al. [46] introduced the concept of semi-variance for uncertain random variables and found the optimal proportions for portfolio selection problems using a mean-semi-variance model. As another down risk measure, Gao et al. [47] optimized portfolio allocation problems using mean-semi-entropy models.

Although the above research discussed portfolio selection problems with different risk measures, there exists some potential areas for investigation. For instance, the concept of the Gini coefficient for uncertain random variables, and consequently mean-Gini-portfolio selection models, should be considered. This paper presents the concept of a partial Gini coefficient for uncertain random variables and investigates its properties. Furthermore, by utilizing inverse uncertainty distributions, we provide a formula for computing the partial Gini coefficient through Monte-Carlo simulation. Moreover, by using the concept of the partial Gini coefficient for uncertain random variables, we present a mean-partial-Gini coefficient with uncertain random returns. For a better understanding, we provide some numerical examples to display the performance of the mean-partial Gini portfolio selection model. The main contributions of this paper can be found in the following aspects. We investigate a mean-partial Gini portfolio selection model via considering security returns as uncertain random variables and convert the mean-partial Gini model into an equivalent ordinary programming model. By assigning performance ratio and diversification indices (using the Rosenbluth, Herfindahl, and comprehensive concentration indices), we compare the mean-partial Gini model with the traditional mean-variance Markowitz model using a Wilcoxon non-parametric test.

This paper is organized as follows: Some preliminaries concepts, definitions and theorems related to uncertain variables and uncertain random variables are provided in Section 2. Section 3 presents the concept of the partial Gini coefficient of uncertain random variables and investigates a formula for computing the partial Gini coefficient using inverse uncertainty distributions. Section 4 applies the partial Gini coefficient to portfolio selection problems and provides some computational examples to display the importance of the partial Gini coefficient in the case of portfolio selection models. Finally, some concluding remarks and potential works for future research are provided in Section 5.

2. Preliminaries

In this section, we provide some preliminaries about chance theory and, consequently, for special case uncertainty theory.

In many situations, randomness and uncertainty appear together in complex phenomena. In order to explain these situations, Liu [23] presented chance theory as an extension of uncertainty theory.

Suppose $(\Gamma ,\mathcal{L},\mathcal{M})$, and $(\Omega ,\mathcal{A},Pr)$ are the uncertainty and probability space, respectively. The product $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},Pr)$ is said to be a chance space. Any element $\Theta \in \mathcal{L}\times \mathcal{A}$ is said to be an event in the chance space.

Definition 1 

(Liu [23]). The chance measure of event Θ is defined as

$$\begin{array}{c}\hfill \mathrm{Ch}\{\Theta \}={\int }_0^{1}Pr\left\{\omega \in \Omega |\mathcal{M}\left\{\gamma \in \Gamma \left|\right(\gamma ,\omega )\in \Theta \right\}\ge x\right\}dx.\end{array}$$

Definition 2 

(Liu [23]). An uncertain random variable is a measurable function ξ from a chance space $(\Gamma ,\mathcal{L},\mathcal{M})\times (\Omega ,\mathcal{A},Pr)$ to the set of real numbers, i.e., $\{\xi \in B\}$ is an event in $\mathcal{L}$ × $\mathcal{A}$ for any Borel set B of real numbers.

In order to measure an uncertain random variable, Liu [23] presented the concept of chance distribution, as follows:

Definition 3 

(Liu [23]). Suppose ξ is an uncertain random variable. Then, its chance distribution is defined by

$$\begin{array}{c}\hfill \Phi \left(x\right)=\mathrm{Ch}\{\xi \le x\},\forall x\in \mathbb{R}.\end{array}$$

Theorem 1 

(Liu [23]). Let ${\eta }_1,{\eta }_2,\dots ,{\eta }_m$ be independent random variables with probability distributions ${\Psi }_1,{\Psi }_2,\dots ,{\Psi }_m$, and let ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ be independent uncertain variables with uncertainty distributions ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n$, respectively. If f is a measurable function, then the uncertain random variable

$$\xi =f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$$

has a chance distribution

$$\begin{array}{c}\hfill \Phi \left(x\right)={\int }_{{\mathbb{R}}^m}F(x,y_1,y_2,\dots ,y_m)d{\Psi }_1\left(y_1\right)d{\Psi }_2\left(y_2\right)\dots d{\Psi }_m\left(y_m\right),\end{array}$$

where

$$F(x,y_1,y_2,\dots ,y_m)=\mathcal{M}\{f(y_1,y_2,\dots ,y_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)\le x\}$$

is the uncertainty distribution of $f(y_1,y_2,\dots ,y_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$ for any real number $y_1,y_2,\dots ,$ $y_m,$ and is determined by ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n.$

Remark 1. 

The chance distribution can be explained as an expectation with respect to the probability distribution functions.

$$\begin{array}{cc}\hfill \Phi \left(x\right)& ={\int }_{{\mathbb{R}}^m}F(x,y_1,y_2,\dots ,y_m)d{\Psi }_1\left(y_1\right)d{\Psi }_2\left(y_2\right)\dots d{\Psi }_m\left(y_m\right)\hfill \\ \hfill & =E_{Pr}\left[F(x,{\eta }_1,{\eta }_2,\dots ,{\eta }_m)\right].\hfill \end{array}$$

Definition 4 

(Liu [23]). Let ξ be an uncertain random variable. Then, its expected value can be defined by

$$\begin{array}{c}\hfill E\left[\xi \right]={\int }_0^{\infty }\mathrm{Ch}\{\xi \ge x\}dx-{\int }_{-\infty }^0\mathrm{Ch}\{\xi <x\}dx,\end{array}$$

where at least one of the two integrals is finite.

Theorem 2 

(Liu [23]). Suppose ${\eta }_1,{\eta }_2,\dots ,{\eta }_m$ are independent random variables with probability distributions ${\Psi }_1,{\Psi }_2,\dots ,{\Psi }_m$, respectively, and suppose ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ are independent uncertain variables with uncertainty distributions ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n$, respectively. If f is a measurable function, then

$$\xi =f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$$

has an expected value

$$\begin{array}{cc}\hfill E\left[\xi \right]& ={\int }_{{\mathbb{R}}^m}E\left[f(y_1,y_2,\dots ,y_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)\right]d{\Psi }_1\left(y_1\right)\dots d{\Psi }_m\left(y_m\right)\hfill \\ \hfill & =E_{Pr}\left[E_{Un}\left[f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)\right]\right].\hfill \end{array}$$

Theorem 3 

(Liu [23]). Suppose ${\eta }_1,{\eta }_2,\dots ,{\eta }_m$ are independent random variables with probability distributions ${\Psi }_1,{\Psi }_2,\dots ,{\Psi }_m$, respectively, and suppose ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ are independent uncertain variables with uncertainty distributions ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n$, respectively. If $f({\eta }_1,\dots ,{\eta }_m,{\tau }_1,\dots ,{\tau }_n)$ is a continuous and strictly increasing function (or strictly decreasing function) with respect to ${\tau }_1,\dots ,{\tau }_n$, then,

$$\xi =f({\eta }_1,{\eta }_2,\dots ,{\eta }_m,{\tau }_1,{\tau }_2,\dots ,{\tau }_n)$$

has an expected value

$$\begin{array}{cc}\hfill E\left[\xi \right]& ={\int }_{{\mathbb{R}}^m}{\int }_0^1{F}^{-1}(\alpha ,y_1,y_2,\dots ,y_m)d\alpha d{\Psi }_1\left(y_1\right)\dots d{\Psi }_m\left(y_m\right)\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^m}{\int }_0^{1}f({\Phi }_1^{-1}\left(\alpha \right),\dots ,{\Phi }_n^{-1}\left(\alpha \right),y_1,y_2,\dots ,y_m)d\alpha d{\Psi }_1\left(y_1\right)\dots d{\Psi }_m\left(y_m\right).\hfill \end{array}$$

In order to characterize the association between two uncertain random variables, Ahmadzade and Gao [34] proposed the concept of covariance, as follows:

Definition 5 

(Ahmadzade and Gao [34]). Let ${\xi }_1$ and ${\xi }_2$ be two uncertain random variables. Then, the covariance of ${\xi }_1$ and ${\xi }_2$ is defined by

$$\begin{array}{c}\hfill Cov({\xi }_1,{\xi }_2)=E\left[({\xi }_1-E\left[{\xi }_1\right])({\xi }_2-E\left[{\xi }_2\right])\right].\end{array}$$

Theorem 4 

(Ahmadzade and Gao [34]). Let ${\tau }_1$ and ${\tau }_2$ be independent uncertain variables with the uncertainty distributions ${\Phi }_1$ and ${\Phi }_2$, respectively. In addition, let ${\eta }_1$ and ${\eta }_2$ be independent random variables with probability distribution functions ${\Psi }_1$ and ${\Psi }_2$, respectively. Then, for ${\xi }_1=f_1({\eta }_1,{\tau }_1)$ and ${\xi }_2=f_2({\eta }_2,{\tau }_2)$, we have

$$\begin{array}{c}\hfill Cov({\xi }_1,{\xi }_2)={\int }_{{\mathbb{R}}^2}{\int }_0^1\left(F_1^{-1}(\alpha ,y_1)-E\left[{\xi }_1\right]\right)\left(F_2^{-1}(\alpha ,y_2)-E\left[{\xi }_2\right]\right)d\alpha d{\Psi }_1\left(y_1\right)d{\Psi }_2\left(y_2\right).\end{array}$$

In particular, for two independent uncertain variables ${\tau }_1$ and ${\tau }_2$, we have

$$\begin{array}{c}\hfill Cov({\tau }_1,{\tau }_2)={\int }_0^1\left({\Phi }_1^{-1}\left(\alpha \right)-E\left[{\tau }_1\right]\right)\left({\Phi }_2^{-1}\left(\alpha \right)-E\left[{\tau }_2\right]\right)d\alpha .\end{array}$$

For more details about the covariance of uncertain variables, see Zhao et al. [17].

Theorem 5 

(Ahmadzade and Gao [34]). Let ${\tau }_1,{\tau }_2,\dots ,{\tau }_n$ be independent uncertain variables, and let ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n$ be independent random variables with probability distribution functions ${\eta }_1,{\eta }_2,\dots ,{\eta }_n$, respectively. Suppose ${\xi }_1=f_1({\tau }_1,{\eta }_1),{\xi }_2=f_2({\tau }_2,{\eta }_2),\dots ,{\xi }_n=f_n({\tau }_n,{\eta }_n)$. Then,

$$\begin{array}{c}\hfill Var({\xi }_1+{\xi }_2+\dots +{\xi }_n)=\sum _{i=1}^{n}Var\left({\xi }_i\right)+2\sum _{i=1}^n\sum _{j=i+1}^{n}Cov({\xi }_i,{\xi }_j).\end{array}$$

In particular, if uncertain random variables reduce to uncertain variables, the results in Zhao et al. [17] are confirmed.In fact, we have

$$\begin{array}{c}\hfill Var({\tau }_1+\dots +{\tau }_n)=\sum _{i=1}^{n}Var\left({\tau }_i\right)+2\sum _{i=1}^n\sum _{j=i+1}^{n}Cov({\tau }_i,{\tau }_j).\end{array}$$

3. Partial Gini Coefficient for Uncertain Random Variables

In this section, we propose the concept of the partial Gini coefficient for uncertain random variables. By employing an inverse uncertainty distribution, we obtain an equation for the computing partial Gini coefficient of uncertain random variables. To begin with, we explore the concept of the Gini coefficient for uncertain variables which was proposed by Gao et al. [18].

Definition 6 

(Gao et al. [18]). Suppose τ is an uncertain variable with uncertainty distribution Φ. The Gini coefficient of the uncertain variable τ is defined by

$$\begin{array}{c}\hfill GC\left(\tau \right)=2Cov(\tau ,\Phi (\tau \left)\right).\end{array}$$

Theorem 6 

(Gao et al. [18]). Let τ be an uncertain variable with uncertainty distributions Φ. Then τ has the Gini coefficient

$$\begin{array}{c}\hfill GC\left(\tau \right)=2{\int }_0^1\alpha \left({\Phi }^{-1}\left(\alpha \right)-\mu \right)d\alpha .\end{array}$$

As an extension of the Gini coefficient for uncertain variables, we present the concept of a Gini coefficient for uncertain random variables, as follows:

Definition 7. 

Suppose ξ is an uncertain random variable with chance distribution Φ. The Gini coefficient of uncertain variable ξ is defined by

$$\begin{array}{c}\hfill GC\left(\xi \right)=2Cov(\xi ,\Phi (\xi \left)\right).\end{array}$$

However, one question may arise. How much of the Gini coefficient of the uncertain random variable is associated with the uncertain variable? To solve this problem, we introduce the concept of the partial Gini coefficient.

Definition 8. 

Suppose that η is a random variable and τ is an uncertain variable, while $\xi =f(\eta ,\tau )$ is an uncertain random variable. The partial Gini of the uncertain random variable ξ is defined as follows:

$$\begin{array}{c}\hfill PGC\left(\xi \right)=2{\int }_{\mathbb{R}}Cov\left(f(y,\tau ),F(\tau ,y)\right)d\Psi \left(y\right)\end{array}$$

where $F(x,y)$ is the uncertainty distribution of the uncertain variable $f(y,\tau )$ for any real number y.

Remark 2. 

It has been mentioned that several concepts in chance theory, such as chance distribution, expected value, and partial entropy, can be written as an expectation with respect to the probability distribution function. Using this fact, we present the concept of the partial Gini coefficient as an expectation with respect to the probability distribution functions, as follows:

$$\begin{array}{cc}\hfill PGC\left(\xi \right)& =2{\int }_{\mathbb{R}}Cov\left(f(y,\tau ),F(\tau ,y)\right)d\Psi \left(y\right)\hfill \\ \hfill & =2E_{Pr}\left[Cov\left(f(\eta ,\tau ),F(\tau ,\eta )\right)\right].\hfill \end{array}$$

Remark 3. 

If an uncertain random variable reduces to an uncertain one, then the partial Gini coefficient becomes the Gini coefficient of uncertain variables (Definition 6).

Theorem 7. 

Suppose that η is a random variable and τ is an uncertain variable, while $\xi =f(\eta ,\tau )$ is an uncertain random variable. The partial Gini of the uncertain random variable ξ is

$$\begin{array}{c}\hfill PGC\left(\xi \right)=2{\int }_{{\mathbb{R}}^m}{\int }_0^1\alpha \left(F^{-1}(\alpha ,y)-\mu \right)d\alpha d\Psi \left(y\right).\end{array}$$

Proof of Theorem 7. 

It is clear that the inverse uncertainty distribution of $f(y,\tau )$ is equal to $F^{-1}(\alpha ,y)$. Moreover, by using operational law, the inverse uncertainty distribution of $F(\tau ,y)$ is equal to $\alpha$. In addition, Theorem 3 implies the expected value of the uncertain variable $F(\tau ,y)$ is

$$\begin{array}{cc}\hfill E\left[F(\tau ,y)\right]& ={\int }_0^{1}F\left(F^{-1}(\alpha ,y),y\right)\mathrm{d}\alpha \hfill \\ \hfill & ={\int }_0^1\alpha \mathrm{d}\alpha =\frac{1}{2}.\hfill \end{array}$$

The definition of the partial Gini coefficient and Theorem 4 imply that

$$\begin{array}{cc}\hfill PGC\left(\xi \right)& =2{\int }_{\mathbb{R}}Cov\left(f(y,\tau ),F(\tau ,y)\right)\mathrm{d}\Psi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{\int }_0^1\left(F^{-1}(\alpha ,y)-E\left[f(y,\tau )\right]\right)\left(\alpha -\frac{1}{2}\right)\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{\int }_0^1\left(F^{-1}(\alpha ,y)-E\left[f(y,\tau )\right]\right)\alpha \mathrm{d}\alpha \mathrm{d}\Psi \left(y\right)\hfill \\ \hfill & -{\int }_{\mathbb{R}}{\int }_0^1\left(F^{-1}(\alpha ,y)-E\left[f(y,\tau )\right]\right)\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F^{-1}(\alpha ,y)-E\left[f(y,\tau )\right]\right)\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F^{-1}(\alpha ,y)-\mu \right)\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right),\hfill \end{array}$$

the latter equality is affirmed by Theorem 3, i.e.,

$$\begin{array}{c}\hfill \mu =E\left[\xi \right]={\int }_{\mathbb{R}}E\left[f(y,\tau )\right]\mathrm{d}\Psi \left(y\right).\end{array}$$

□

Remark 4. 

If an uncertain random variable is converted into an uncertain one, the above theorem becomes Theorem 6.

Remark 5. 

By using Theorem 6, we can calculate the partial Gini coefficient of uncertain random variables using Monte-Carlo simulation. In fact, the Gini coefficient can be explained as follows:

$$\begin{array}{c}\hfill PGC\left(\xi \right)=E_{Pr}\left[U\left(F^{-1}(U,\eta )-\mu \right)\right],\end{array}$$

where $U,\eta$ are random variables with a standard uniform distribution and $\Psi \left(y\right)$, respectively. Then, we apply Monte-Carlo simulation as follows:

First, we simulate $u_1,u_2,\dots ,u_N$ and $y_1,y_2,\dots ,y_M$ from the uniform and probability distribution $\Psi \left(y\right)$.

Second, we compute $u_i\left(F^{-1}(u_i,y_j)-\mu \right)$ for $i=1,2,\dots ,N$ and $j=1,2,\dots ,M$.

Third, by applying a strong law of large numbers, we approximate a partial Gini coefficient via $\frac{1}{NM}{\sum }_{j=1}^M{\sum }_{i=1}^N{u}_i\left(F^{-1}(u_i,y_j)-\mu \right)$.

Theorem 8. 

Let η be a random variable with the probability distribution function Ψ. In addition, let τ be an uncertain variable with the uncertainty distribution Φ. Consider

$$\xi =f(\eta ,\tau )$$

as an uncertain random variable. Then, we have

$$\begin{array}{c}\hfill PGC\left(\xi \right)\le 2\sqrt{\frac{Var\left(\xi \right)}{3}}.\end{array}$$

Proof. 

By using Theorem 6 and Cauchy-Schwarz inequality, we have

$$\begin{array}{ccc}\hfill PGC\left(\xi \right)& =& 2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F^{-1}(\alpha ,y)-\mu \right)\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right)\hfill \\ & \le & 2\sqrt{{\int }_{\mathbb{R}}{\int }_0^1{(F^{-1}(\alpha ,y)-\mu )}^2\mathrm{d}\alpha \mathrm{d}\Psi \left(y\right){\int }_0^1{\alpha }^2\mathrm{d}\alpha }\hfill \\ & =& 2\sqrt{\frac{Var\left(\xi \right)}{3}}.\hfill \end{array}$$

□

Example 1. 

Consider $\eta \sim Gamma(10,5)$ as a random variable. Moreover, suppose $\tau \sim \mathcal{N}(0,1)$ is a normal uncertain variable. Set $\xi ={\eta }^3\tau$ as an uncertain random variable. Then, partial Gini coefficient of ξ is

$$\begin{array}{c}\hfill PGC\left(\xi \right)=2{\int }_0^{\infty }{\int }_0^1\alpha y^3\left(\frac{\sqrt{3}}{\pi }ln\frac{\alpha }{1-\alpha }\right)\frac{5^{10}{y}^9{e}^{-5y}}{\Gamma \left(10\right)}d\alpha dy=5.822033.\end{array}$$

Example 2. 

Consider $\eta \sim Gamma(15,5)$ as a random variable. Suppose $\tau \sim \mathcal{LOGN}(0,1)$ is a normal uncertain variable. Set $\xi =\frac{\tau }{{\eta }^4}$ as an uncertain random variable. Then, the partial Gini coefficient of ξ is

$$\begin{array}{c}\hfill PGC\left(\xi \right)=2{\int }_0^{\infty }{\int }_0^1\frac{\alpha }{y^4}·exp\left(\frac{\sqrt{3}}{\pi }ln\frac{\alpha }{1-\alpha }\right)\frac{5^{15}{y}^{14}{e}^{-5y}}{\Gamma \left(15\right)}d\alpha dy=0.030041.\end{array}$$

Example 3. 

Let $\eta \sim U(a,b)$ be a uniform random variable. In addition, suppose $\tau \sim L(c,d)$ is a linear uncertain variable. Consider $\xi =\eta \tau$ as an uncertain random variable. Then, the partial Gini coefficient of ξ is

$$\begin{array}{ccc}\hfill PGC\left(\xi \right)& =& 2{\int }_c^d{\int }_0^1\alpha \left(y\left(c(1-\alpha )+d\alpha \right)-\frac{c+d}{2}\frac{a+b}{2}\right)\frac{1}{b-a}d\alpha dy\hfill \\ & =& \frac{a+b}{2}\times \frac{d-c}{6}.\hfill \end{array}$$

Theorem 9. 

Suppose η is a random variable with the probability distribution function Ψ. Moreover, suppose τ is an uncertain variable with uncertainty distribution function Φ. Consider $\xi =\tau +\eta$ an uncertain random variable. Then, we have $PGC\left(\xi \right)=GC\left(\tau \right)$.

Proof of Theorem 9. 

It is clear that $F^{-1}(\alpha ,y)=y+{\Phi }^{-1}\left(\alpha \right)$ and $E\left[\xi \right]=E\left[\tau \right]+E\left[\eta \right]$. By employing Theorem 7, we have

$$\begin{array}{cc}\hfill PGC\left(\xi \right)& =2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F^{-1}(\alpha ,y)-E\left[\xi \right]\right)\mathrm{d}\alpha \mathrm{d}\Phi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(y+{\Phi }^{-1}\left(\alpha \right)-E\left[\tau \right]-E\left[\eta \right]\right)\mathrm{d}\alpha \mathrm{d}\Phi \left(y\right)\hfill \\ \hfill & =2{\int }_{\mathbb{R}}\left(y-E\left[\eta \right]\right)\mathrm{d}\Psi \left(y\right){\int }_0^1\alpha \mathrm{d}\alpha +2{\int }_0^1\alpha \left({\Phi }^{-1}\left(\alpha \right)-E\left[\tau \right]\right)\mathrm{d}\alpha \hfill \\ \hfill & =0+2{\int }_0^1\alpha \left({\Phi }^{-1}\left(\alpha \right)-E\left[\tau \right]\right)\mathrm{d}\alpha \hfill \\ \hfill & =2{\int }_0^1\alpha \left({\Phi }^{-1}\left(\alpha \right)-E\left[\tau \right]\right)\mathrm{d}\alpha \hfill \\ \hfill & =GC\left(\tau \right).\hfill \end{array}$$

□

Theorem 10. 

Suppose ${\eta }_1$ and ${\eta }_2$ are independent random variables with the probability distribution functions ${\Psi }_1$ and ${\Psi }_2$, respectively. In addition, ${\tau }_1$ and ${\tau }_2$ are independent uncertain variables with the uncertainty distributions ${\Phi }_1$ and ${\Phi }_2$, respectively. Furthermore, consider ${\xi }_1=f_1({\eta }_1,{\tau }_1)$ and ${\xi }_2=f_2({\eta }_2,{\tau }_2)$ two uncertain random variables. Then, we have

$$PGC({\xi }_1+{\xi }_2)=PGC\left({\xi }_1\right)+PGC\left({\xi }_2\right).$$

Proof of Theorem 10. 

The linearity property of the expected value for uncertain random variables implies that

$$\begin{array}{c}\hfill E[{\xi }_1+{\xi }_2]=E\left[{\xi }_1\right]+E\left[{\xi }_2\right].\end{array}$$

In addition, it is clear that the inverse uncertainty distribution of ${\xi }_1+{\xi }_2$ is equal to $F_1^{-1}(\alpha ,y_1)+F_2^{-1}(\alpha ,y_2)$. Then, Theorem 7 shows that

$$\begin{array}{ccc}\hfill PGC({\xi }_1+{\xi }_2)& =& 2{\int }_{{\mathbb{R}}^2}{\int }_0^1\alpha \left(F_1^{-1}(\alpha ,y_1)+F_2^{-1}(\alpha ,y_2)-\mu \right)\mathrm{d}\alpha \mathrm{d}{\Psi }_1\left(y_1\right)\mathrm{d}{\Psi }_2\left(y_2\right)\hfill \\ & =& 2{\int }_{{\mathbb{R}}^2}{\int }_0^1\alpha \left({\Phi }_1^{-1}\left(\alpha \right)+{\Phi }_2^{-1}\left(\alpha \right)-E\left[{\tau }_1\right]-E\left[{\tau }_2\right]\right)\mathrm{d}\alpha \mathrm{d}{\Psi }_1\left(y_1\right)\mathrm{d}{\Psi }_2\left(y_2\right)\hfill \\ & =& 2{\int }_{{\mathbb{R}}^2}{\int }_0^1\alpha \left(F_1^{-1}(\alpha ,y_1)-E\left[{\xi }_1\right]+F_2^{-1}(\alpha ,y_2)-E\left[{\xi }_2\right]\right)\mathrm{d}\alpha \mathrm{d}{\Psi }_1\left(y_1\right)\mathrm{d}{\Psi }_2\left(y_2\right)\hfill \\ & =& 2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F_1^{-1}(\alpha ,y_1)-E\left[{\xi }_1\right]\right)\mathrm{d}\alpha \mathrm{d}{\Psi }_1\left(y_1\right)\hfill \\ & & +2{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F_2^{-1}(\alpha ,y_2)-E\left[{\xi }_2\right]\right)\mathrm{d}\alpha \mathrm{d}{\Psi }_2\left(y_2\right)\hfill \\ & =& PGC\left({\xi }_1\right)+PGC\left({\xi }_2\right).\hfill \end{array}$$

□

4. Portfolio Selection of Uncertain Returns

In this section, we present a new uncertain random portfolio allocation model based on the concept of the partial Gini coefficient. In many situations, historical stocks and newly listed ones are encountered together. For old stocks, we consider the empirical distribution function as an estimation of the probability distribution function based on historical frequency. For newly listed stocks, we assign the degree of belief of experts to the uncertainty distribution of the stocks.

In the traditional Markowitz model, the expected value and variance are assigned to investment returns and risk, respectively. However, other indices such as entropy, partial entropy, semi-entropy, and semi-variance have been considered as a risk measure in portfolio optimization. On the other hand, as evidence, Gao et al. [34] showed that the Gini coefficient is more general than variance as an efficient device to measure risk.Considering the above research, we obtain a mean-partial Gini optimization model for uncertain random returns by considering the partial Gini coefficient as a risk measure.

In many situations, we encounter n securities including a historical term with random returns ${\eta }_i$ and a rare term with uncertain returns ${\tau }_i$. Thus, we have n securities ${\xi }_i={\tau }_i+{\eta }_i,i=1,2,\dots ,n.$ In addition, consider $p_i$ the investment proportion in security $i=1,2,\dots ,n.$ Then, the return of total investment is an uncertain random variable $p_1{\xi }_1+p_2{\xi }_2+\dots +p_n{\xi }_n$. Now, we can explain the uncertain random portfolio selection model by minimizing the partial Gini coefficient as a risk measure, as follows:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle \underset{p\in {\mathbb{R}}^n}{min}PGC(p_1{\xi }_1+p_2{\xi }_2+\dots +p_n{\xi }_n),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ E[p_1{\xi }_1+p_2{\xi }_2+\dots +p_n{\xi }_n]\ge \lambda ,\hfill \\ \\ p_1+p_2+...+p_n=1,p_i\ge 0,i=1,2,\dots ,n.\hfill \end{array}\right.}$$

(1)

where E is the expected value and $PGC$ denotes the partial Gini coefficient, and $\lambda$ represents the minimum expectation level.

Suppose ${\eta }_1,{\eta }_2,\dots ,{\eta }_n$ are independent random variables with probability distribution functions ${\Psi }_1,{\Psi }_2,\dots ,{\Psi }_n$, and ${\tau }_1,{\tau }_2,\dots ,{\eta }_n$ are independent uncertain variables with uncertainty distributions ${\Phi }_1,{\Phi }_2,\dots ,{\Phi }_n$, respectively. We assign ${\xi }_i=f_i({\eta }_i,{\tau }_i)$, $i=1,2,\dots ,n$ as an uncertain random return and $F_i^{-1}(\alpha ,y_i)$ as the inverse uncertainty distribution of the uncertain variable $f_i(y_i,{\tau }_i)$. Using Theorem 6, we convert the partial Gini objective function to

$$\begin{array}{c}\hfill \sum _{i=1}^n{p}_i{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F_i^{-1}(\alpha ,y_i)-\mu \right)\mathrm{d}\alpha \mathrm{d}{\Psi }_i\left(y_i\right).\end{array}$$

Moreover, using Theorem 3, the expected value constraint is as follows:

$$\begin{array}{c}\hfill \sum _{i=1}^n{x}_i{\int }_{\mathbb{R}}{\int }_0^1{F}_i^{-1}(\alpha ,y_i)\mathrm{d}\alpha \mathrm{d}{\Psi }_i\left(y_i\right)\le \lambda .\end{array}$$

Then, we explain Model (1) based on the following equivalent model:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle \underset{p\in {\mathbb{R}}^n}{min}\sum _{i=1}^n{p}_i{\int }_{\mathbb{R}}{\int }_0^1\alpha \left(F_i^{-1}(\alpha ,y_i)-\mu \right)\mathrm{d}\alpha \mathrm{d}{\Psi }_i\left(y_i\right),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ \sum _{i=1}^n{p}_i{\int }_{\mathbb{R}}{\int }_0^1{F}_i^{-1}(\alpha ,y_i)\mathrm{d}\alpha \mathrm{d}{\Psi }_i\left(y_i\right)\ge \lambda ,\hfill \\ \\ \sum _{i=1}^n{p}_i=1,p_i\ge 0,i=1,2,\dots ,n.\hfill \end{array}\right.}$$

(2)

It should be mentioned that Model (2) is a linear program, where we can obtain the optimal solution directly via LINGO, MATLAB, and so on.

In order to compare the mean-partial Gini model with another model, we selected the mean-variance primary model based on Markowitz’s approach. Ahmadzade and Gao [34] presented this mean-variance model, as follows:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle \underset{p\in {\mathbb{R}}^n}{min}Var(p_1{\xi }_1+P_2{\xi }_2+\dots +p_n{\xi }_n),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ E[p_1{\xi }_1+p_2{\xi }_2+\dots +p_n{\xi }_n]\ge \lambda ,\hfill \\ \\ \sum _{i=1}^n{p}_i=1,p_i\ge 0,i=1,2,\dots ,n.\hfill \end{array}\right.}$$

(3)

where $Var$ is the variance, E denotes the expected value, and $\lambda$ represents the lower bound for the expected value. In addition, Model (3) can be changed to the following model by means of Theorem 5.

$${\displaystyle \left\{\begin{array}{c}{\displaystyle \underset{p\in {\mathbb{R}}^n}{min}\sum _{i=1}^n{p}_i^{2}Var\left(\xi \right)+\sum \sum _{i\ne j}{p}_i{p}_{j}Cov({\xi }_i,{\xi }_j),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ p_{1}E\left[{\xi }_1\right]+p_{2}E\left[{\xi }_2\right]+\dots +p_{n}E\left[{\xi }_n\right]\ge \lambda ,\hfill \\ \\ \sum _{i=1}^n{p}_i=1,p_i\ge 0,i=1,2,\dots ,n.\hfill \end{array}\right.}$$

(4)

It is clear that Model (4) is a quadratic program, where we can obtain its optimal solution directly via LINGO, MATLAB, and so on.

In generally, the optimization algorithm can be explained as follows:

1.

Simulate N data from the determined distribution (e.g., uniform, normal, exponential);

2.

Simulate M data from the uniform distribution on unit interval;

3.

Consider the sample average as an approximation for the quantities, such as expected value, variance, and partial Gini coefficient;

4.

Convert the portfolio selection models to ordinary programming models;

5.

Apply LINGO or MATLAB software to calculate the optimal proportions of the portfolio selection models.

In order to compare the mean-partial Gini model with other models, we need to recall several concentration indices and performance ratios. Moreover, consequently, we explain the diversification indices corresponding to the concentration indices. The first index is the Herfindahl, with most application to economic concentration, for more details, see [48]. The index is

$$\begin{array}{c}\hfill HI=\sum _{i=1}^n{p}_i^2,\end{array}$$

where $p_i$ is the proportion of investment in security $i,$$i=1,2,\dots ,n.$

The second is the Resenbluth index, which is expressed as weighted sums, as follows:

$$\begin{array}{c}\hfill RI={\left(2\sum _{i=1}^{n}i{p}_i\right)}^{-1},\end{array}$$

where $p_i,$$i=1,2,\dots ,n$ are the investment proportions in the securities, see Rosenbluth [49].

Finally, the comprehensive concentration [50] index is as follows:

$$\begin{array}{c}\hfill CCI=p_1+\sum _{i=2}^n{p}_i^2\left(1+(1-p_i)\right),\end{array}$$

where $p_i,$$i=1,2,\dots ,n$ are arranged in non-increasing order, and consequently, $p_1$ is the largest.

Using the results of Woerheide and Persson [48], we consider the Herfindahl index, Rosenbluth index, and comprehensive concentration index to measure the degree of diversification, as follows:

$$\begin{array}{cc}\hfill CHI& =1-\sum _{i=1}^n{p}_i^2,\hfill \\ \hfill CRI& =1-{\left(2\sum _{i=1}^{n}i{p}_i\right)}^{-1},\hfill \\ \hfill CCCI& =1-p_1-\sum _{i=2}^n{p}_i^2\left(1+(1-p_i)\right).\hfill \end{array}$$

Based on the above indices, investors prefer a model with a large value of diversification index. As another characterization measure, performance ratio was applied to compare the approaches of the different selection models, for more details, see [51,52,53]. The performance ratio was as follows:

$$\begin{array}{c}\hfill PR=\frac{E\left[{\sum }_{i=1}^n{p}_i{\xi }_i\right]}{\sqrt{Var({\sum }_{i=1}^n{p}_i{\xi }_i})}.\end{array}$$

Example 4. 

Based on the empirical distribution function of securities in the old markets and the degree of belief of experts in the newly listed securities, suppose the investor selects 4 securities from different companies with normal uncertainty distributions and uniform probability distribution functions, as in Ahmadzade and Gao [34]. In this example, 4 securities are supposed as uncertain random variables with ${\xi }_i={\eta }_i+{\tau }_i,$ $i=1,2,3,4.$ Random and uncertain terms are displayed in

Table 1. Uncertain Random Securities.

No	Uncertain Terms	Random Terms
1	${\tau }_1\sim \mathcal{N}(100,15)$	${\eta }_1\sim U(108,132)$
2	${\tau }_2\sim \mathcal{N}(115,13)$	${\eta }_2\sim U(165,195)$
3	${\tau }_3\sim \mathcal{N}(125,14)$	${\eta }_3\sim U(240,260)$
4	${\tau }_4\sim \mathcal{N}(130,20)$	${\eta }_4\sim U(162,188)$

.

By means of Theorem 6 and Monte-Carlo simulation, we calculated the partial Gini coefficient for all securities. Then, we optimize the mean-partial Gini model the same as Model (2), directly. The optimal proportions under the mean-partial Gini model with different values of λ are shown in

Table 2. Optimal Values in Model (2).

$\mathit{\lambda }$	$({\mathit{x}}_1,{\mathit{x}}_2,{\mathit{x}}_3,{\mathit{x}}_4)$	Objective Partial Gini
290	(0, 1, 0, 0)	7.167276
297	(0, 0.975, 0.025, 0)	7.187059
300	(0, 0.9375, 0.0625, 0)	7.201734
310	(0, 0.8125, 0.1875, 0)	7.270650
320	(0, 0.6875, 0.3125, 0)	7.339566
330	(0, 0.5625, 0.4375, 0)	7.408482
350	(0, 0.3125, 0.8125, 0)	7.546314
360	(0, 0.1875, 0.8125, 0)	7.615230
370	(0, 0.0625, 0.9375, 0)	7.684147
375	(0, 0, 1, 0)	7.718605

.

In order to compare the mean-entropy model with the traditional mean-variance model, Chen et al. [54] applied several diversification indices with respect to a fixed maximum entropy, such as $\gamma =150$. They preferred the mean-entropy model with large diversification indices, rather than mean-variance model. In fact, their approach was not reasonable. Since, they made the decision based on the value of γ.

For the same fixed λ in Models (2) and (4), we derived optimal proportions and their related performance ratio and diversification indices. Then, we iterated this approach for several values of λ in Models (2) and (4). Therefore, we set the pairs $(CHIMV,CHIMPG)$, $(CRIMV,CRIMPG)$, $(CCCIMV,CCCIMPG)$ and $(PRMV,PRMPG)$, where $CHIMV,$ $CRIMV,$ $CCCIMV$ and $PRMV$ are related indices in the mean-variance(MV) model, and similarly, $CHIMPG$, $CRIMPG,$ $CCCIMPG$, and $PRMPG$ are related indices in the mean-partial Gini (MPG) model. Since, the mean-variance and mean-partial Gini model were applied to the same dateset, we had a pair or related samples. In order to recognize the significance difference between the performance ratio and diversification indices in the mean-variance and mean-partial Gini model, we used a Wilcoxon non-parametric test in SPSS software. The null hypothesis in the Wilcoxon test was the median of the population of differences between the paired data being zero. The alternative hypothesis was that it is not true.In fact, the null and alternative hypotheses in the Wilcoxon test can be expressed as follows: $H_0$ means there is no significant difference between two related samples. Similarly, $H_1$ means there is significant difference between two related samples.

Table 3. Comparative analysis of difference between the four indices in Models (2) and (4).

Test Statistics	PRMPG-PRMV	CHIMPG-CHIMV	CRIMPG-CRIMV	CCCIMPG-CCCIMV
Z	−1.172	−0.817	−1.572	−0.1433
Asymp.Sig. (2-tailed)	0.241	0.414	0.116	0.152

shows that the null hypotheses of comparing related indices between the mean-variance and mean-partial Gini model were accepted at the significance level $0.05$. This means there was no significant difference between related indices in the mean-variance and mean-partial Gini model.

Example 5. 

Considering the following proportions of the candidate securities

$$w_1,w_2,\dots ,w_n,z_1,z_2,\dots ,z_m,$$

the total return can be formulated as

$$\begin{array}{c}\hfill \xi =\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j,\end{array}$$

where ${\sum }_{i=1}^n{w}_i{\tau }_i$ and ${\sum }_{j=1}^m{z}_j{\eta }_j$ are the total returns corresponding to newly listed and historical securities, respectively. It is clear that ξ is an uncertain random variable as the total return. Suppose we have ten historical securities with the descriptive statistics displayed in

Table 4. Descriptive Statistics.

Security	Mean	Gini
1	0.023	0.98
2	0.092	0.711
3	0.027	0.625
4	0.326	1.371
5	0.021	0.921
6	0.026	0.614
7	0.016	0.654
8	0.170	0.645
9	−0.198	1.15
10	−0.198	0.742

. Moreover, we have five newly listed securities with the uncertainty distributions in

Table 5. Uncertain Securities.

No	Uncertain Terms
1	${\tau }_1\sim \mathcal{L}(0.02,0.091)$
2	${\tau }_2\sim \mathcal{L}(-0.003,0.012)$
3	${\tau }_3\sim \mathcal{L}(0.06,0.14)$
4	${\tau }_4\sim \mathcal{L}(0.08,0.19)$
5	${\tau }_5\sim \mathcal{L}(0.009,0.21)$

.

In order to compare the performance and diversification of the mean-Gini model with the mean-variance model, we considered two minimax models as follows:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle minPGC(\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j),}\hfill \\ \\ {\displaystyle maxE[\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j],}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,...,n,j=1,2,...,m.\hfill \end{array}\right.}$$

(5)

and

$${\displaystyle \left\{\begin{array}{c}{\displaystyle minVar(\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j),}\hfill \\ \\ {\displaystyle maxE[\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j],}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,...,n,j=1,2,...,m.\hfill \end{array}\right.}$$

(6)

The above models can be converted into equivalent programming models as follows:

$${\displaystyle \left\{\begin{array}{c}{\displaystyle min\sum _{i=1}^n{w}_{i}PGC\left({\tau }_i\right)+\sum _{j=1}^m{z}_{j}PGC\left({\eta }_j\right),}\hfill \\ \\ {\displaystyle max\sum _{i=1}^n{w}_{i}E\left[{\tau }_i\right]+\sum _{j=1}^m{z}_{j}E\left[{\eta }_j\right],}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,...,n,j=1,2,...,m.\hfill \end{array}\right.}$$

(7)

and

$${\displaystyle \left\{\begin{array}{c}{\displaystyle min\sum _{i=1}^n{w}_i^{2}Var\left({\tau }_i\right)+\sum _{k=1}^n\sum _{l=1}^n{w}_k{w}_{l}Cov({\tau }_k,{\tau }_l)+\sum _{j=1}^m{z}_j^{2}Var\left({\eta }_j\right)+\sum _{k=1}^m\sum _{l=1}^m{z}_k{z}_{l}Cov({\eta }_k,{\eta }_l),}\hfill \\ \\ {\displaystyle max\sum _{i=1}^n{w}_{i}E\left[{\tau }_i\right]+\sum _{j=1}^m{z}_{j}E\left[{\eta }_j\right],}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,...,n,j=1,2,...,m.\hfill \end{array}\right.}$$

(8)

By applying Models (7) and (8) to the portfolio selection, we have linear and quadratic programming models, respectively. By solving the models, we obtained the performance ratio and diversification indices in

Table 6. Descriptive Statistics.

Model	PR	CHI	CRI	CCCI
MPG	0.499	0.5220	0.6082	0.181
MV	0.480	0.3555	0.5442	0.111

.

The results in Table 6 prove that the mean-partial Gini model was better than the mean-variance model with respect to performance ratio and diversification indices. For a better illustration, we consider the following models under a constrained mean for fixed λ.

$${\displaystyle \left\{\begin{array}{c}{\displaystyle minPGC(\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ {\displaystyle E[\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j]\ge \lambda ,}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,\dots ,n,j=1,2,\dots ,m.\hfill \end{array}\right.}$$

(9)

and

$${\displaystyle \left\{\begin{array}{c}{\displaystyle minVar(\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j),}\hfill \\ \\ {\displaystyle subjectto:}\hfill \\ \\ {\displaystyle E[\sum _{i=1}^n{w}_i{\tau }_i+\sum _{j=1}^m{z}_j{\eta }_j]\ge \lambda ,}\hfill \\ \\ \sum _{i=1}^n{w}_i+\sum _{j=1}^m{z}_j=1w_i\ge 0,z_j\ge 0i=1,2,\dots ,n,j=1,2,\dots ,m.\hfill \end{array}\right.}$$

(10)

For the same fixed λ in Models (9) and (10), we obtained the optimal proportions and their related performance ratio and diversification indices. Then, we iterated this approach for several values of λ in Models (9) and (10). Therefore, we set the pairs $(CHIMV,CHIMPG)$, $(CRIMV,CRIMPG)$, $(CCCIMV,CCCIMPG)$ and $(PRMV,PRMPG)$, where $CHIMV$, $CRIMV$, $CCCIMV$, and $PRMV$ are related indices in the mean-variance (MV) model, and similarly, $CHIMPG$, $CRIMPG$, $CCCIMPG$, and $PRMPG$ are related indices in the mean-partial Gini (MPG) model. Since, the mean-variance and mean-partial Gini model were applied to the same dataset, we had a pair or related samples.

Table 7. Comparative analysis of difference between the four indices in Models (9) and (10).

Test Statistics	PRMPG-PRMV	CHIMPG-CHIMV	CRIMPG-CRIMV	CCCIMPG-CCCIMV
Z	−1.377	−0.357	−0.459	−0.874
Asymp.Sig. (2-tailed)	0.169	0.721	0.646	0.382

shows that the null hypotheses of comparing related indices between the mean-variance and mean-partial Gini models were accepted at the significance level $0.05$. This means there was no significant difference between the related indices in the mean-variance and mean-partial Gini models.

5. Conclusions

In this paper, we presented the partial Gini coefficient for uncertain random variables and applied it to portfolio allocation problems via considering partial Gini coefficient for uncertain random variables and investigated its properties via inverse uncertainty distributions. In order to display the application of the mean-partial Gini portfolio selection models, several computational examples were provided. Furthermore, by applying the performance ratio and several types of diversification indices to the optimal proportions of the portfolio selection, we compared the mean-partial Gini model with the mean-variance model. In general, the mean-partial Gini model was better than the mean-variance model.

This paper contributes to investigating the portfolio selection problem by considering partial Gini coefficient as a risk measure for uncertain random returns. The main contributions were in three areas: First, we presented the concept of the partial Gini coefficient for uncertain random variables and investigated its properties via inverse uncertainty distributions. Second, we used the partial Gini coefficient for the portfolio allocation problem and obtained a mean-partial Gini model. Finally, for better illustration of the main results of the mean-partial Gini model, we provided some computational examples and compared the mean-partial Gini model with the mean-variance model, with respect to the performance ratio and diversification indices.

Some potential further works should be considered. First, we plan to establish some extended types of Gini coefficient, and their mathematical properties will be discussed. Second, to obtain more diversified proportions in portfolio selection problems, we will add some type of entropy to the mean-partial Gini portfolio selection models. Finally, considering several types of down risk measure, such as semi-entropy and semi-variance, in mean-partial Gini portfolio selection models with uncertain random returns may be a potential area for future research.