Covariance Representations and Coherent Measures for Some Entropies

We obtain covariance and Choquet integral representations for some entropies and give upper bounds of those entropies. The coherent properties of those entropies are discussed. Furthermore, we propose tail-based cumulative residual Tsallis entropy of order α (TCRTE) and tail-based right-tail deviation (TRTD); then, we define a shortfall of cumulative residual Tsallis (CRTES) and shortfall of right-tail deviation entropy (RTDS) and provide some equivalent results. As illustrated examples, the CRTESs of elliptical, inverse Gaussian, gamma and beta distributions are simulated.

To estimate and identify risk measure, the law invariance (A) is an essential requirement.When a risk measure ρ further satisfies (A1)-(A4), then ρ is said to be coherent.It is well known that value-at-risk (VaR) and expected shortfall (ES) are the two extremely important risk measures used in banking and insurance.The VaR and ES at confidence level p ∈ (0, 1) for a random variable (r.v.) V with cumulative distribution function (cdf) F V are defined as VaR p (V) = F −1 V (p): = inf{p : F V (x) ≥ p}, and respectively.If F V is continuous, then ES equals the tail conditional expectation (TCE), which is written as where v p = VaR p (V).
To capture the variability of the risk V beyond the quantile v p , Furman and Landsman [4] proposed the tail standard deviation (TSD) risk measure where p ∈ (0, 1), λ ≥ 0 denotes the loading parameter and SD p (V) the tail standard deviation measure defined as SD p (V) = TV p (V).
Here, TV p (V) = E[(V − TCE p (V)) 2 |V > v p ] is the tail variance of V.As its extension, Furman et al. [5] introduced the Gini shortfall (GS), which is defined by where TGini p (V) = 2 (1−p) 2   1 p F −1 V (s)(2s − (1 + p))ds is tail-Gini functional.Recently, Hu and Chen [6] further proposed a shortfall of cumulative residual entropy (CRE), defined by where TCRE p (V) = Here, h v p (s) = 0 for s ∈ [0, p), and h v p (s) = 1−s 1−p log 1−s 1−p for s ∈ [p, 1].Inspired by those works, our main motivation is to find coherent shortfalls of entropy, which is the generalization of TSD, GS and CRES.These shortfalls of entropy can be used to capture the variability of a financial position.For specific financial applications, we can refer to [5,7,8].To this aim, we give covariance and Choquet integral representations for some entropies, and provide upper bounds of those entropies.These representations not only make it easier for us to judge their cohesiveness, but also facilitate the extension of these results to two-dimensional and multi-dimensional cases in the future.Furthermore, we define TCRTE and TRTD, and propose CRTES and RTDS.As illustrated examples, CRTESs of elliptical, inverse Gaussian, gamma and beta distributions are simulated.
The remainder of this paper is structured as follows.Section 2 provides the covariance and Choquet integral representations for some entropies.Section 3 introduces some tail-based entropies.In Section 4, we propose two shortfalls of entropy, and give some equivalent results.The CRTESs of some parametric distributions are presented in Section 5. Finally, Section 6 concludes this paper and summarizes two possible research studies in the future.
Throughout the paper, let (Ω, F , P) be an atomless probability space.For a random variable V with cumulative distribution function (cdf) F V , we use U V to denote any uniform [0, 1] random variable such that be the set of all random variables on (Ω, F , P) with a finite kth-moment.Denote by L 0 + the set of all non-negative random variables.g denotes the first derivative of g.Notation v + = max{v, 0}, and 1 A (•) is the indicator function of set A.

Covariance and Choquet Integral Representations for Some Entropies
In this section, we derive covariance and Choquet integral representations for some entropies, which include initial, weighted and dynamic entropies.In addition, the upper bounds of these entropies are established.
Given g defined in [0, 1] with g(0) = g(1) = 0, weighted function ψ(•) and a r.v.X with cdf F X , the initial and weighted entropies (forms) are defined as, respectively, To derive the covariance of entropy, we first introduce below lemma.

Initial Entropy
To find the covariance represent of initial entropy, we give the following theorem.
Theorem 1.Let g be a continuous and almost everywhere differentiable function in [0, 1] with l = g (a.e.) and g(0) = g(1) = 0. Further, there exists a unique minimum (or maximum) point t 0 ∈ (0, 1) such that g is decreasing on [0, t 0 ] and increasing on [t 0 , 1] (or there exists t 0 ∈ (0, 1) such that g is increasing on [0, t 0 ] and decreasing on [t 0 , 1]).Suppose that X ∈ L 2 and l ∈ L 2 .Then, we have (5) Proof.Since g(u) is almost everywhere differentiable in [0, 1], and g(0) = g(1) = 0, there exists a unique minimum (or maximum) point t 0 ∈ (0, 1).Hence, we can use g to induce Lebesgue-Stieltjes measures on the Borel-measurable spaces ([0, where we have used Fubini's theorem in the third equality.Further, using Lemma 1, we obtain Remark 1.Note that the function g is of bounded variation since g has the following representation g = g 1 + g 2 , where g 1 is increasing and g 2 is decreasing.Similar results can be found in Lemma 3 of [9].However, the result of this article is different from Lemma 3 of [9].The integral interval and integrand are different, with one integrand being a function of F (i.e., g(F)) and the other being a function of F (i.e., g( F)).So, our result cannot be obtained from theirs.

Weighted Entropy
Weighted entropy is an extension of initial entropy, which is an initial entropy associated with a weight function.We have the corresponding theorem as follows.
Proof.Similar to the proof of Theorem 1, we have where we have used Fubini's theorem in the third equality.Note that and E[l(U X )] = 1 0 l(u)du = g(1) − g(0) = 0. Therefore, we obtain ending the proof.

Dynamic (Weighted) Entropy
Dynamic entropy is also a generalization of an initial entropy that focuses on feasible choices of the ranges (upper tail or lower tail).
where g t (v) = 0 for v ∈ [0, F X (t)), and g t Corollary 5. Let g be a concave function in [0, 1] with l = g (a.e.) and g(0) = g(1) = 0. Suppose that X ∈ L 2 and l ∈ L 2 .Then, we have The distribution function of a random variable X (t) = [X|X ≤ t] can be written as F X (t) , when x ≤ t, 0, otherwise.
In particular, when X ∈ L 0 + , the above measure is the cumulative residual Tsallis entropy of order α introduced by Rajesh and Sunoj [13].Note that when α → 1, it reduces to cumulative entropy (E (X)), defined as (see [14]) In particular, when X ∈ (0, c), the above measure is called the fractional cumulative residual entropy of X by Xiong et al. [15].
In particular, when X ∈ L 0 + , the above measure is the right-tail deviation introduced by Wang [16].
In particular, when X ∈ L 0 + , the above measure is the extended Gini coefficient (see [7]).As a special case, when r = 2, the extended Gini coefficient becomes the simple Gini (see [5]).
In particular, when X ∈ (0, c), the above measure is called the fractional generalized cumulative residual entropy of X by Di Crescenzo et al. [17].
In particular, if α is a positive integer, say Then, FGRE α (X) identifies with the generalized cumulative residual entropy (GCRE n (X)) that has been introduced by Psarrakos and Navarro [18], i.e., Further, In particular, when X ∈ (0, c), the above measure is called the fractional generalized cumulative entropy of X by Di Crescenzo et al. ([17]).
where g t (v) = 0 for v ∈ [0, F X (t)), and g t (v) = in Corollary 5, Equation ( 12) denoted by DCRT α,t (X); we have In particular, when X ∈ L 0 + , the above measure is the dynamic cumulative residual Tsallis entropy of order α introduced by Rajesh and Sunoj [13].
In particular, when X ∈ L 0 + and α = 2, we obtain (dynamic Gini mean semi-difference) where g t (v) = 0 for v ∈ [0, F X (t)), and g t (v) = where g t (v) = 0 for v ∈ [0, F X (t)), and g t In particular, when X ∈ L 0 + , the above measure denotes the dynamic generalized cumulative residual entropy introduced by Psarrakos and Navarro [18].
. Then, we have In particular, when X ∈ L 0 + , the above measure is the dynamic WGCRE introduced by Tahmasebi [25].

Discussion
Note that the above entropy risk measures satisfy (B1) standardization by their covariance representations.For any c ∈ R, using F −1 X+c (u) = F −1 X (u) + c, we obtain that initial entropy and simple dynamic entropy risk measures satisfy (B2) location invariance, but weighted entropy risk measures do not satisfy (B2).Therefore, initial entropy and simple dynamic entropy risk measures are measures of variability.
When g : [0, 1] → R is finite variation and g(1) = g(0) = 0, the signed Choquet integral is defined by When g is right-continuous, then Equation ( 20) can be expressed as Furthermore, when g is absolutely continuous, with dg(s) = l(s)ds, then Equation ( 21) can be expressed as From ( 21), we can see that the signed Choquet integral satisfies the co-monotonically additive property ([32]).Thus, initial entropy and simple dynamic entropy risk measures are co-monotonically additive measures of variability.
These initial entropy risk measures can be applied to the predictability of the failure time of a system (see [11,14]).The weighted entropy risk measures are shift-dependent measures of uncertainty, and can be applied to some practical situations of reliability and neurobiology (see [35,36]).The dynamic entropy risk measures can be used to capture effects of the age t of an individual or an item under study on the information about the residual lifetime (see [29]).
The initial, weighted and dynamic entropy risk measures are closely related, as shown in Figure 1: From a risk measure point of view, the initial entropy risk measures can capture the variability of a financial position as a whole.The dynamic entropy risk measures can depict the variability of a financial position focused on feasible choices of the ranges (upper tail or lower tail).
In finance and risk management, Markowitz's mean-variance portfolio theory plays a vital role in modern portfolio theory.It is known that the initial entropy and simple dynamic entropy risk measures are measures of variability.We can replace variance with the initial entropy and simple dynamic entropy risk measures, respectively.The initial entropy risk measure is used to capture ordinary (general) risk, and it is favored by investors, such as the firm's ordinary business and the shareholders' interests.The dynamic entropy risk measure is used to depict the tails of risks (extreme events), which is to reduce (or avoid) the impact of extreme events and is favored by regulators and decision makers (see [37]).For example, we give CRTES λ α,p (X) for different distributions in Section 5, and also use the R software to compute CRTES λ α,p (X) for p ∈ [0.9, 1), shown in Figures 2-8.When α → 1, CRTES λ α,p (X) reduces to TCRE p (X) given by Hu and Chen [6], we can observe the difference between our results and Hu and Chen's results through Figures 2-8.Other potential applications of these entropy risk measures need to be further explored in the future.Fig. 2: 19 2 with λ = 0.5; (c) CRTES λ α,p (X) of λ = 0, λ = 0.5 and λ = 1 with α = 1 2 .Fig. 4: 22

Shortfalls of Entropy
We now introduce two risk measures of entropy shortfall, which are linear combinations of ES p , TCRTE α,p and TRTD p , respectively: where p ∈ [0, 1) is also a confidence level, and λ ≥ 0 is a loading parameter Theorem 5. Assume that p ∈ (0, 1), λ ∈ [0, ∞) and the convex cone X = L 2 .

Elliptical Distributions
Consider an elliptical random variable X ∼ E 1 (µ, σ 2 , g 1 ).If the probability density function (pdf) of X exists, its form will be (see [40]) where µ and σ > 0 are location and scale parameters, respectively.Moreover, g 1 (t), t ≥ 0, is the density generator of X, and is denoted by X ∼ E 1 (µ, σ 2 , g 1 ).The density generator g 1 satisfies the condition and the normalizing constant c 1 is given by Cumulative generator G 1 (u) and normalizing constant c * 1 are, respectively, defined as follows: Landsman and Valdez [40] proved that where w p = x p −µ σ .Now, several important cases, including normal, Student-t, logistic and Laplace distributions, are given as follows.
Example 21. (Normal distribution) Let X ∼ N 1 (µ, σ 2 ).In this case, the density generators are written as and the normalization constants are given by c Then, where w p = x p −µ σ .
Without loss of generality [for the convenience of simulation], let X ∼ N 1 (0, 1); by Equations ( 29), ( 30) and (34), we can use the R software to compute TCRTE α,p (X) and CRTES λ α,p (X) for p ∈ [0.9, 1), and the results are shown in Figure 2. From Figure 2a, we find that when α is fixed, TCRTE α,p is decreasing in p.Moreover, when p is fixed, TCRTE α,p is also decreasing in α.As we can see in Figure 2b, when α is fixed, CRTES λ α,p is increasing in p, while CRTES λ α,p will be decreasing in α when p is fixed.In Figure 2c, we observe that when λ is fixed, CRTES λ α,p is increasing in p.Moreover, when p is fixed, TCRTE α,p is also increasing in λ.
The normalization constants are given by , and where Γ(•) and B(•, •) denote gamma and beta functions, respectively.Then, where w p = x p −µ σ .
From Figure 3a, we find that the degree of freedom, m, has a great impact on TCRTE α,p .TCRTE α,p is decreasing in m.When m is small, TCRTE α,p is increasing in p, while TCRTE α,p will be decreasing in p instead of increasing when m is larger than a threshold.From Figure 3b, we find that when α is fixed, TCRTE α,p is increasing in p.However, when p is fixed, TCRTE α,p is decreasing in α.In Figure 3c, we observe that when α is fixed, CRTES λ α,p is increasing in p.However, when p is fixed, CRTES λ α,p is decreasing in α.From Figure 3d, we find that when λ is fixed, CRTES λ α,p is increasing in p.Moreover, when p is fixed, CRTES λ α,p is also increasing in λ.
It is seen from Figure 4a that the α has a little impact on the values of TCRTE α,p .For fixed p, TCRTE α,p is decreasing in α.From Figure 4b, we observe that when α is fixed, CRTES λ α,p is increasing in p.However, when p is fixed, CRTES λ α,p is decreasing in α.In Figure 4c, we find that when λ is fixed, CRTES λ α,p is increasing in p.Moreover, when p is fixed, CRTES λ α,p is also increasing in λ.
It is seen from Figure 5a that p has almost no impact on TCRTE α,p .For fixed α, TCRTE α,p is almost the same in p.However, α has a great impact on TCRTE α,p .For fixed p, TCRTE α,p is decreasing in α.In Figure 5b, we observe that when α is fixed, CRTES λ α,p is increasing in p.However, when p is fixed, CRTES λ α,p is decreasing in α.From Figure 5c, we find that when λ is fixed, CRTES λ α,p is increasing in p.Moreover, when p is fixed, CRTES λ α,p is also increasing in λ.Let X ∼ IG(10, 1); by Equations ( 29), ( 30) and ( 38), we can use the R software to compute TCRTE α,p (X) and CRTES λ α,p (X) for p ∈ [0.9, 1), and the results are shown in Figure 6.

Figure 1 .
Figure 1.The relationship between three entropy risk measures.