# Notes on Cumulative Entropy as a Risk Measure

Saeid Tahmasebi 1  and Hojat Parsa 2
• 1 Department of Statistics, Persian Gulf University, Bushehr, Iran
• 2 Department of Economics, Persian Gulf University, Bushehr, Iran
Saeid Tahmasebi
and Hojat Parsa

## Abstract

Di Crescenzo and Longobardi [Di Crescenzo and Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 2009, 12, 4072–4087] proposed the cumulative entropy (CE) as an alternative to the differential entropy. They presented an estimator of CE using empirical approach. In this paper, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference for some distributions. We also make empirical comparisons of these measures using samples from stock market in members of the Organization for Economic Co-operation and Development (OECD) countries.

## 1 Introduction and Background

Let X denote the lifetime of a device or a system with probability density function (pdf) f and cumulative distribution function (cdf) F. Then the differential entropy known as Shannon entropy is defined by (see [12])

$H⁢(X)=-∫0+∞f⁢(x)⁢log⁡f⁢(x)⁢d⁢x,$

where, by convention, $0⁢log⁡0=0$. Recently, new measures of information are proposed. Replacing the pdf by the survival function $F¯=1-F$ in Shannon entropy, the cumulative residual entropy (CRE) is defined by Rao et al. [11] as

$ℰ⁢(X)=∫0+∞F¯⁢(x)⁢Λ⁢(x)⁢d⁢x,$

where $Λ⁢(x)=-log⁡F¯⁢(x)$. Asadi and Zohrevand [1] also considered a dynamic version of the CRE. More properties of CRE in residual lifetime are given by Navarro, del Aguila and Asadi [6]. Some new connections between CRE and the residual lifetime are obtained in [5] using the relevation transform. Sordo and Psarrakos [13] provided comparison results for the CRE of systems and their dynamic versions. In a recent paper by Yang [15], it is shown that CRE is an alternative measure of risk for heavy-tailed distributions with infinite variance. Recently, Psarrakos and Navarro [8] introduced an extension of CRE called generalized cumulative residual entropy (GCRE), related to upper records statistics. Psarrakos and Toomaj [9] applied GCRE to problems related to risk measures. Navarro and Psarrakos [7] proved some interesting characterization results based on GCRE. Toomaj, Sunoj and Navarro [14] obtained an interesting connection between CRE and the Gini mean difference as a dispersion measure alternative to standard deviation. A new information measure similar to CRE has been proposed by Di Crescenzo and Longobardi [3] as

$𝒞⁢ℰ⁢(X)=∫0+∞F⁢(x)⁢Λ~⁢(x)⁢d⁢x,$

where $Λ~⁢(x)=-log⁡F⁢(x)$. Note that $0≤𝒞⁢ℰ⁢(X)≤∞$. Properties on CE in past lifetime are given by Di Crescenzo and Longobardi [3] and Navarro, del Aguila and Asadi [6]. Di Crescenzo and Toomaj [4] obtained further results for the CE including stochastic ordering, bounds and characterization results. Motivated by some of the articles mentioned above, in this paper, we aim to investigate some applications of CE as a risk measure analogous to that proposed by Yang [15]. Investors are entering the stock market in order to earn returns or profit, but they are facing a risk of unexpected fluctuation of stock returns. Traditionally, stock market analysts use standard deviation criteria in order to calculate the risk of stock price fluctuation. In this study, we consider a risk measure based on CE and compare it with the standard deviation and the Gini mean difference as applied to the stock market. The paper is organized as follows: Section 2 presents some basic properties of risk measures. We also compare the risk measures $σ⁢(X)$ (standard deviation of X), $𝒞⁢ℰ⁢(X)$ and the Gini mean difference for some distributions. In Section 3, we make empirical comparisons of these risk measures using samples from stock market in OECD countries.

## 2 Some Properties of CE

In this section, we present some basic properties of risk measures. We also compare the risk measure $σ⁢(X)$, $𝒞⁢ℰ⁢(X)$ and the Gini mean difference for some distributions.

Definition 2.1.

Let X be a random variable describing outcomes of a risky asset, and let $𝒳$ be the set of all functions $X:Ω→ℝ$. A mapping $ρ:𝒳→ℝ$ is called a risk measure if it fulfills some conditions as follows:

1. (1)Sub-additivity: If $X,Y∈𝒳$, then $ρ⁢(X+Y)≤ρ⁢(X)+ρ⁢(Y)$.
2. (2)Positive homogeneity: If $X∈𝒳$, $a≥0$, then $ρ⁢(a⁢X)=a⁢ρ⁢(X)$.
3. (3)Monotonicity: If $X≤Y$, then $ρ⁢(X)≤ρ⁢(Y)$.
4. (4)Convexity: If $λ∈[0,1]$ and $X,Y∈𝒳$, then $ρ⁢(λ⁢X+(1-λ)⁢Y)≤λ⁢ρ⁢(X)+(1-λ)⁢ρ⁢(Y)$.
5. (5)Consistency: If $X∈𝒳$, $b∈ℝ$, then $ρ⁢(X+b)=ρ⁢(X)$.

Definition 2.2.

A random variable X is said to be smaller than a random variable Y in decreasing convex order ($X≤dcxY$), if $𝔼⁢(ϕ⁢(X))≤𝔼⁢(ϕ⁢(Y))$ for all decreasing convex functions ϕ such that the expectations exist.

Di Crescenzo and Longobardi [3] obtained the following results for CE. These results include linear transformations, various bounds and stochastic order properties.

Proposition 2.3.

Let X be a random variable with cdf F. Furthermore, let $Y=a⁢X+b$, where $a>0$ and $b≥0$. Then

$𝒞⁢ℰ⁢(Y)=a⁢𝒞⁢ℰ⁢(X).$

Proposition 2.4.

For a non-negative absolutely continuous random variable X with $C⁢E⁢(X)<∞$, we have

$𝒞⁢ℰ⁢(X)=𝔼⁢[T(2)⁢(X)],$

where

$T(2)(X):=-∫x+∞logF(z)dz,x≥0.$

Note that $T(2)⁢(⋅)$ is a decreasing convex function.

Proposition 2.5.

Let X and Y be two non-negative random variables such that $X≤dcxY$. Then

$𝒞⁢ℰ⁢(X)≤𝒞⁢ℰ⁢(Y).$

Proposition 2.6.

For a non-negative absolutely continuous random variable X with $E⁢(X)<∞$, we have

$𝒞⁢ℰ⁢(X)≤𝔼⁢(X).$

Proposition 2.7.

Cumulative entropy satisfies convexity of a risk measure.

Proof.

For all $λ∈[0,1]$ and $X,Y∈𝒳$, according to the sub-additivity of cumulative entropy, we have

$𝒞⁢ℰ⁢(λ⁢X+(1-λ)⁢Y)≤𝒞⁢ℰ⁢(λ⁢X)+𝒞⁢ℰ⁢((1-λ)⁢Y).$

Now, according to the positive homogeneity of cumulative entropy, we get

$𝒞⁢ℰ⁢(λ⁢X)+𝒞⁢ℰ⁢((1-λ)⁢Y)=λ⁢𝒞⁢ℰ⁢(X)+(1-λ)⁢𝒞⁢ℰ⁢(Y).$

Hence, recalling (2.1) and (2.2), the result follows. ∎

Definition 2.8.

A celebrated measure of income inequality, the Gini index, is defined as

$Gini⁡(X)=DG⁢(X)2⁢𝔼⁢(X),$

where $DG(X):=∫0+∞2F¯(x)(1-F¯(x))dx$ is the Gini mean difference as a dispersion measure.

Proposition 2.9.

If X is a non-negative absolutely continuous random variable with cdf F, then

$𝒞⁢ℰ⁢(X)≥DG*⁢(X),$

where $DG*⁢(X)=DG⁢(X)2$.

Proof.

The proof is similar to that of [3, Proposition 4.3]. ∎

The standard deviation $σ⁢(X)$ of a risk X is a common risk measure in insurance. Ramsay [10] showed that the standard deviation is not an appropriate tool in measuring large insurance risks with long-tailed skewed distributions. In the following examples, we compare the risk measures $σ⁢(X)$, $𝒞⁢ℰ⁢(X)$ and $DG*⁢(X)$ for some well-known distributions. Throughout these examples, we use the ratio

$η⁢(X)=𝒞⁢ℰ⁢(X)σ⁢(X).$

Example 2.1.

If X is a random variable with cdf $FX⁢(x)=1-exp⁡(-θ⁢x)$, $x>0$, then

$η⁢(X)=π26-1.$

Note that $DG*⁢(X)≤𝒞⁢ℰ⁢(X)<σ⁢(X)$.

Example 2.2.

If X is a random variable with cdf $FX⁢(x)=xa$, $0, then

$η⁢(X)=32.$

Note that $DG*⁢(X)≤𝒞⁢ℰ⁢(X)<σ⁢(X)$.

Example 2.3.

If X has an inverse Weibull distribution with cdf $FX⁢(x)=exp⁡{-c/xγ}$, $x>0$, $c>0$, $γ>0$, then

$η⁢(X)=Γ⁢(1-1γ)γ⁢[Γ⁢(2-1γ)-Γ2⁢(1-1γ)]12.$

Note that for $γ>3$, we have $DG*⁢(X)≤𝒞⁢ℰ⁢(X)<σ⁢(X)$.

Example 2.4.

If X has standard normal distribution with pdf $f⁢(x)=12⁢π⁢exp⁡(-x22)$, $-∞, then

$η⁢(X)=𝒞⁢ℰ⁢(X)σ⁢(X)=0.90.$

Note that $DG*⁢(X)≤𝒞⁢ℰ⁢(X)<σ⁢(X)$.

## 3 Empirical Study of Standard Deviation and CE in Stock Market

Let $X1,X2,…,Xm$ be a random sample of size m from an absolutely continuous cdf F. If

$X(1)≤X(2)≤…≤X(m)$

represent the order statistics of the sample $X1,X2,…,Xm$, then the empirical measure of F is defined for $k=1,2,…,m-1$ by

$F^m⁢(x)={0,xX(k+1).$

The empirical measure of $𝒞⁢ℰ⁢(X)$ is defined by Di Crescenzo and Longobardi [3] as

$𝒞⁢ℰ⁢(F^m)=-∑k=1m-1Uk+1⁢(km)⁢log⁡(km),$

where $Uk=X(k)-X(k-1)$, $k=1,2,3,…,m$ and $X(0)=0$. Note that $limm→∞⁡𝒞⁢ℰ⁢(F^m)=𝒞⁢ℰ⁢(X)$. An empirical measure of $DG*⁢(X)$ is obtained by Davidson [2] as

$D^G*⁢(X)=2m2⁢∑k=1mX(k)⁢(k-0.5)-X¯.$

Investors enters the stock market in order to earn returns or profit; however, they are facing a risk of unexpected fluctuation of stock returns. Returns on a stock are combinations of dividend’s and increase in the stock price. Sometimes, investors in stock market face decrease of stock price and loss parts of their properties. Therefore, the major part of the risk is related to stock price fluctuations. Traditionally, stock market analysts use the standard deviation criterion in order to calculate the risk of stock price fluctuation. In this study, we use the annual data of share price index of OECD countries during 2000–2017 to calculate the changes of price share as a part of return on stock and its fluctuation. OECD members are the following:

Australia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Japan, Korea, Luxembourg, Mexico, the Netherlands, New Zealand, Norway, Poland, Portugal, Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, the United Kingdom and the United States.

Table 1 shows the stock market values of standard deviation σ, skewness, kurtosis, CE and $D*$ from OECD countries. Figures 1 and 2 also show the apparent positive correlation between standard deviation and cumulative entropy, which confirms the presence of similar behavior of these two risk measures. From Figure 3, we can conclude that, when the system of volatility (standard deviation) increases, the cumulative entropy increases as well. Likewise, when the system of volatility (standard deviation) has stabilized, cumulative entropy becomes small. Finally, a linear regression equation between σ and CE of share price index in OECD countries is obtained as $𝒞⁢ℰ=0.8106⁢σ$. This model has an $R2$ value 98 %. Similarly, a linear regression equation between σ and $D*$ of share price index in OECD countries is obtained as $D*=0.52808⁢σ$. This model has an $R2$ value 99.7 %.

Table 1

Values of σ, skewness, kurtosis, CE and $DG*⁢(X)$ for stock market in OECD countries.

 Countries σ (%) Skewness Kurtosis $CE⁢(nats)$ $DG*⁢(X)$ AUS 0.1255 -0.42 -0.58 0.1096 0.0687 AUT 0.2224 -0.07 -0.10 0.1892 0.1199 BEL 0.1834 -0.29 -0.80 0.1599 0.1015 CAN 0.1501 0.09 -0.27 0.1259 0.0822 CHE 0.1552 -0.29 -1.32 0.1320 0.0854 CHL 0.1461 0.65 -0.63 0.1097 0.0795 CZE 0.2325 0.27 -0.39 0.1899 0.1275 DEU 0.1968 -0.53 -1.46 0.1641 0.1057 DNK 0.1969 -0.55 -0.88 0.1742 0.1077 ESP 0.1776 -0.07 -1.76 0.1424 0.0976 EST 0.3047 -0.11 0.22 0.2612 0.1638 FIN 0.2995 1.10 3.07 0.2289 0.1519 FRA 0.1873 -0.19 -1.18 0.1551 0.1018 GBR 0.1200 -0.28 -1.22 0.1012 0.0660 GRC 0.2849 0.34 -1.69 0.2116 0.1541 HUN 0.2406 0.19 -0.58 0.1973 0.1330 IRL 0.2166 -1.27 1.12 0.1976 0.1112 ISL 0.3559 -1.33 2.85 0.3230 0.1733 ITA 0.1911 -0.25 -1.42 0.1612 0.1048 JPN 0.2072 0.21 -0.56 0.1696 0.1144 KOR 0.1650 0.05 -1.33 0.1363 0.0908 LUX 0.2623 0.08 -1.04 0.2179 0.1450 MEX 0.1883 0.65 -0.92 0.1397 0.1004 NLD 0.1839 -0.61 -1.27 0.1582 0.0984 NOR 0.2253 0.14 -0.77 0.1877 0.1252 NZL 0.1109 -1.21 1.99 0.1011 0.0562 POL 0.2315 -0.23 -0.94 0.2005 0.1288 PRT 0.1867 0.06 -1.36 0.1529 0.1041 SVK 0.2979 1.82 5.35 0.2059 0.1443 SWE 0.2216 -0.23 -0.61 0.1895 0.1210 TUR 0.4079 1.94 5.38 0.2718 0.1964 USA 0.1306 -0.70 -0.31 0.1185 0.0709

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• Crossref
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R. Davidson, Reliable inference for the Gini index, J. Econometrics 150 (2009), no. 1, 30–40.

• Crossref
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A. Di Crescenzo and M. Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 (2009), no. 12, 4072–4087.

• Crossref
• Export Citation
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A. Di Crescenzo and A. Toomaj, Further results on the generalized cumulative entropy, Kybernetika (Prague) 53 (2017), no. 5, 959–982.

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S. Kapodistria and G. Psarrakos, Some extensions of the residual lifetime and its connection to the cumulative residual entropy, Probab. Engrg. Inform. Sci. 26 (2012), no. 1, 129–146.

• Crossref
• Export Citation
• [6]

J. Navarro, Y. del Aguila and M. Asadi, Some new results on the cumulative residual entropy, J. Statist. Plann. Inference 140 (2010), no. 1, 310–322.

• Crossref
• Export Citation
• [7]

J. Navarro and G. Psarrakos, Characterizations based on generalized cumulative residual entropy functions, Comm. Statist. Theory Methods 46 (2017), no. 3, 1247–1260.

• Crossref
• Export Citation
• [8]

G. Psarrakos and J. Navarro, Generalized cumulative residual entropy and record values, Metrika 76 (2013), no. 5, 623–640.

• Crossref
• Export Citation
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G. Psarrakos and A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math. 309 (2017), 186–199.

• Crossref
• Export Citation
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C. M. Ramsay, Loading gross premiums for risk without using utility theory, Trans. Soc. Actuar. 45 (1993), 305–349.

• [11]

M. Rao, Y. Chen, B. C. Vemuri and F. Wang, Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1220–1228.

• Crossref
• Export Citation
• [12]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–432.

• Crossref
• Export Citation
• [13]

M. A. Sordo and G. Psarrakos, Stochastic comparisons of interfailure times under a relevation replacement policy, J. Appl. Probab. 54 (2017), no. 1, 134–145.

• Crossref
• Export Citation
• [14]

A. Toomaj, S. M. Sunoj and J. Navarro, Some properties of the cumulative residual entropy of coherent and mixed systems, J. Appl. Probab. 54 (2017), no. 2, 379–393.

• Crossref
• Export Citation
• [15]

L. Yang, Study on cumulative residual entropy and variance as risk measure, Fifth International Conference on Business Intelligence and Financial Engineering, IEEE Press, Piscataway (2012), 210–213.

If the inline PDF is not rendering correctly, you can download the PDF file here.

• [1]

M. Asadi and Y. Zohrevand, On the dynamic cumulative residual entropy, J. Statist. Plann. Inference 137 (2007), no. 6, 1931–1941.

• Crossref
• Export Citation
• [2]

R. Davidson, Reliable inference for the Gini index, J. Econometrics 150 (2009), no. 1, 30–40.

• Crossref
• Export Citation
• [3]

A. Di Crescenzo and M. Longobardi, On cumulative entropies, J. Statist. Plann. Inference 139 (2009), no. 12, 4072–4087.

• Crossref
• Export Citation
• [4]

A. Di Crescenzo and A. Toomaj, Further results on the generalized cumulative entropy, Kybernetika (Prague) 53 (2017), no. 5, 959–982.

• [5]

S. Kapodistria and G. Psarrakos, Some extensions of the residual lifetime and its connection to the cumulative residual entropy, Probab. Engrg. Inform. Sci. 26 (2012), no. 1, 129–146.

• Crossref
• Export Citation
• [6]

J. Navarro, Y. del Aguila and M. Asadi, Some new results on the cumulative residual entropy, J. Statist. Plann. Inference 140 (2010), no. 1, 310–322.

• Crossref
• Export Citation
• [7]

J. Navarro and G. Psarrakos, Characterizations based on generalized cumulative residual entropy functions, Comm. Statist. Theory Methods 46 (2017), no. 3, 1247–1260.

• Crossref
• Export Citation
• [8]

G. Psarrakos and J. Navarro, Generalized cumulative residual entropy and record values, Metrika 76 (2013), no. 5, 623–640.

• Crossref
• Export Citation
• [9]

G. Psarrakos and A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math. 309 (2017), 186–199.

• Crossref
• Export Citation
• [10]

C. M. Ramsay, Loading gross premiums for risk without using utility theory, Trans. Soc. Actuar. 45 (1993), 305–349.

• [11]

M. Rao, Y. Chen, B. C. Vemuri and F. Wang, Cumulative residual entropy: A new measure of information, IEEE Trans. Inform. Theory 50 (2004), no. 6, 1220–1228.

• Crossref
• Export Citation
• [12]

C. E. Shannon, A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379–432.

• Crossref
• Export Citation
• [13]

M. A. Sordo and G. Psarrakos, Stochastic comparisons of interfailure times under a relevation replacement policy, J. Appl. Probab. 54 (2017), no. 1, 134–145.

• Crossref
• Export Citation
• [14]

A. Toomaj, S. M. Sunoj and J. Navarro, Some properties of the cumulative residual entropy of coherent and mixed systems, J. Appl. Probab. 54 (2017), no. 2, 379–393.

• Crossref
• Export Citation
• [15]

L. Yang, Study on cumulative residual entropy and variance as risk measure, Fifth International Conference on Business Intelligence and Financial Engineering, IEEE Press, Piscataway (2012), 210–213.

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