# Stein Type Lemmas for Location-Scale Mixture of Generalized Skew-Elliptical Random Vectors

Authors: Baishuai Zuo; Chuancun Yin
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

Inspired by the work of Adcock, Landsman, and Shushi (2019) which established the Stein’s lemma for generalized skew-elliptical random vectors, we derive Stein type lemmas for location-scale mixture of generalized skew-elliptical random vectors. Some special cases such as the location-scale mixture of elliptical random vectors, the location-scale mixture of generalized skew-normal random vectors, and the location-scale mixture of normal random vectors are also considered. As an application in risk theory, we give a result for optimal portfolio selection.


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## Body

## 1. Introduction and Motivation

Since Stein [1] provides an expression EhXX−μ for normal random variable X, where hx is an almost differentiable function, and a number of scholars have generalized the formula. For example, Landsman [2] gives Stein’s lemma for 2-dimensional elliptical distributions; Landsman and Nešlehová [3] and Landsman et al. [4] derive Stein’s lemma for multivariate elliptical distributions; Landsman et al. [5] establish Stein-type inequality for symmetric generalized hyperbolic distributions; Adcock et al. [6] derive Stein’s lemma for generalized skew-elliptical distributions. The result has been applied in statistics, insurance, and finance. For example, Landsman et al. [5] and Landsman et al. [7] apply this lemma in risk theory.In the study by Kim and Kim [8], the class of normal mean-variance mixture distributions is introduced. The random vector X is said to be an n-dimensional normal mean-variance mixture variable if X=μ+Θγ+Θ1/2AZ, where Z∼Nk0,Ik, the k-dimensional normal random vectors with the identity covariance matrix; A is an n×k matrix; Θ is a scalar random variable that follows a nonnegative distribution with the density πθ, independent of Z; and the following are sontant vectors in Rn:(1)μ=μ1,μ2,…,μnT,γ=γ1,γ2,…,γnT.These specification implies that conditionally, XΘ=θ∼Nnμ+θγ,θΣ, where Σ=AAT. Inspired by this, we consider a class of location-scale mixture of generalized skew-elliptical distributions, which is generalization of the class of normal mean-variance mixture distributions. In this paper, we generalize Stein’s lemma by Adcock et al. [6] to the case of location-scale mixture of generalized skew-elliptical random vectors.The rest of the paper is organized as follows. Section2 introduces the definitions and properties of the location-scale mixture of generalized skew-elliptical distributions. In Section 3, we derive three Stein-type lemmas. In Section 4, we give several special cases. An optimal portfolio selection (a three-fund theorem) for location-scale mixture of generalized skew-elliptical random vectors is given in Section 5.


## 2. Mixture of Generalized Skew-Elliptical Distributions

In this section, we introduce the class of location-scale mixture of generalized skew-elliptical (LSMGSE) distributions and some of its properties.LetY be an n-dimensional generalized skew-elliptical random vector and denoted by Y∼GSEnμ,Σ,gn,π⋅. If its probability density function exists, the form will be (see [6])(2)fYy=2Σgn12y−μTΣ−1y−μπΣ−1/2y−μ,y∈ℝn,where(3)fXx:=1Σgn12x−μTΣ−1x−μ,x∈ℝn,is the density of n-dimensional elliptical random vector X∼Enμ,Σ,gn. Here, μ is an n×1 location vector, Σ is an n×n scale matrix, and gnu, u≥0, is the density generator of X. πx,x∈ℝn, is called the skewing function satisfying π−x=1−πx and 0≤πx≤1. The characteristic function of X takes the form φXt=expitTμψ1/2tTΣt,t∈ℝn, with function ψt:0,∞⟶ℝ, called the characteristic generator (see [9]). Suppose A be an n×n matrix and b be an n×1 vector. Then,(4)AY+b∼GSEnAμ+b,ATΣA,gn,π⋅.To establish Stein’s lemma forn-dimensional generalized skew-elliptical distributions, we use the cumulative generator G¯nu. It takes the following form (see [7] or [10]):(5)G¯nu=∫u∞gnvdv.LetX∗∼Enμ,Σ,G¯n be an elliptical random vector with generator G¯nu, whose density function (if it exists) is(6)fX∗x=−1ψ′0ΣG¯n12x−μTΣ−1x−μ,x∈ℝn.LetY∗∼GSEnμ,Σ,G¯n,π⋅ be a generalized skew-elliptical random vector.We callZ∼LSMGSEnμ,Σ,β,Θ,gn,π⋅ as an n-dimensional LSMGSE distribution with location parameter μ, positive definite scale matrix Σ, and skew function π⋅, if(7)Z=μ+Θβ+Θ1/2Σ1/2Y,where β∈ℝn and Y∼GSEn0,In,gn,π⋅. Assume that Y is independent of nonnegative scalar random variable Θ. We have(8)ZΘ=θ∼GSEnμ+θβ,θΣ,gn,π⋅.


## 3. Main Result

In this section, we consider a random vector(9)Z∼LSMGSEnμ,Σ,β,Θ,gn,π⋅,with location parameter μ, positive definite scale matrix Σ, and skew function π⋅ as (7).Letϖ:ℝm⟶ℝ,1≤m≤n, be an almost everywhere differentiable function, and we write(10)∇ϖz1=∂ϖz1∂z1,∂ϖz1∂z2,…,∂ϖz1∂znT.We derive a Stein-type lemma for location-scale mixture of generalized skew-elliptical random vectors below. PartitionZ=Z1T,Z2TT, where Z1=Z1,Z2,…,ZmT and Z2=Zm+1,Zm+2,…,ZnT. μ=μ1T,μ2TT and μ=μ1T,μ2T are also of similar partition.Theorem 1.
LetZ∼LSMGSEnμ,Σ,β,Θ,gn,π⋅ be an n-dimensional location-scale mixture of generalized skew-elliptical random vector defined as (7). Assume that function ϖ satisfies E∇1ϖS1∗<∞. Then,(11)EϖZ1Z−μ=EθCovME∇ϖS1∗+2EϖM1∗∇πθΣ−1/2M∗−μ−θβ+EθEϖS1θβ,where(12)S=S1T,S2TT=ZΘ=θ∼GSEnμ+θβ,θΣ,gn,π⋅,M=M1T,M2TT∼Enμ+θβ,θΣ,gn,S∗=S1∗T,S2∗TT∼GSEnμ+θβ,θΣ,G¯n,π⋅,M∗=M1∗T,M2∗TT∼Enμ+θβ,θΣ,G¯n.Proof.
Using tower property of expectations, we obtain(13)EϖZ1Z−μ=EΘEϖZ1Z−μΘ.while(14)EϖZ1Z−μΘ=θ=EϖZ1Θ=θZ−μΘ=θ=EϖZ1Θ=θZΘ=θ−μ+θβ+θβ=EϖZ1Θ=θZΘ=θ−μ+θβ+EϖZ1|Θ=θθβ=EϖS1S−μ+θβ+EϖS1θβ=CovME∇ϖS1∗+2EϖM1∗∇πθΣ−1/2M∗−μ−θβ+EϖS1θβ,where the last equality have used (4), (8), and Theorem 3 by Adcock et al. [6]. Therefore, we obtain (11), which completes the proof of Theorem 1.Remark 1.
From formula (8), we find that EϖZ1Z−μΘ is a special case of Theorem 3 by Adcock et al. [6].
The following theorems give two special forms of Stein-type lemmas for location-scale mixture of generalized skew-elliptical random vectors.Theorem 2.
LetZ∼LSMGSEnμ,Σ,β,α,gn,π⋅ be an n-dimensional location-scale mixture of generalized skew-elliptical random vector with(15)Z=μ+V−1β+V−1/2Σ1/2Y,where V∼betaα,1. Assume the function ϖ satisfies E∇1ϖS1∗<∞. Then,(16)EϖZ1Z−μ=EvCovME∇ϖS1∗+2EϖM1∗∇πv−1Σ−1/2M∗−μ−v−1β+EvEϖS1v−1β,where(17)S=S1T,S2TT=ZV=v∼GSEnμ+v−1β,v−1Σ,gn,π⋅,M=M1T,M2TT∼Enμ+v−1β,v−1Σ,gn,S∗=S1∗T,S2∗TT∼GSEnμ+v−1β,v−1Σ,G¯n,π⋅,M∗=M1∗T,M2∗TT∼Enμ+v−1β,v−1Σ,G¯n.Proof.
LettingΘ=V−1 in Theorem 1, we directly obtain (16). This completes the proof of Theorem 2.Remark 2.
Lettingπ⋅=1/2 in Theorem 2, we obtain a Stein-type lemma for location-scale mixture of elliptical random vectors:(18)EϖZ1Z−μ=EvCovME∇ϖM1∗+EvEϖM1v−1β.Theorem 3.
LetZ∼LSMGSEnμ,Σ,β,Θ,gn,π⋅ be an n-dimensional location-scale mixture of generalized skew-elliptical random vector defined as (7). Assume the function ϖ satisfies E∇1ϖS∗<∞. Then,(19)EϖZZ−μ=EθCovME∇ϖS∗+2EϖM∗∇πθΣ−1/2M∗−μ−θβ+EθEϖSθβ.Proof.
Lettingϖz1=ϖz in Theorem 1, we obtain (19). This completes the proof of Theorem 3.Remark 3.
Lettingπ⋅=1/2 in Theorem 3, we obtain a Stein-type lemma for location-scale mixture of elliptical random vectors:(20)EϖZZ−μ=EθCovME∇ϖM∗+EθEϖMθβ.


## 4. Special Cases

In this section, we consider several special cases including the location-scale mixture of elliptical distribution, the location-scale mixture of generalized skew-normal distribution, the location-scale mixture of skew-normal distribution, and the location-scale mixture of normal distribution.Example 1.
Lettingπ⋅=1/2 in Theorem 1, Stein-type lemma for location-scale mixture of elliptical random vector is given by(21)EϖZ1Z−μ=EθCovME∇ϖM1∗+EθEϖM1θβ.Remark 4.
We find that (21) can be regarded as a special analogue case of Vanduffel and Yao [11].Example 2.
SupposeY∼GSNnμ,Σ,π⋅ is an n-dimensional generalized skew-normal random vector with probability density function (pdf) as follows:(22)fYy=2Σ2πn/2exp−12y−μTΣ−1y−μπγTΣ−1/2y−μ,y∈ℝn, where γ=γ1,γ2,…,γnT and function π⋅:ℝ⟶ℝ. Letting G¯nu=gnu=2π−n/2exp−u and(23)πΣ−1/2y−μ=πγTΣ−1/2y−μ,in Theorem 1. Assuming that function ϖ satisfies E∇1ϖS1<∞, Stein-type lemma for location-scale mixture of generalized skew-normal random vector Z∼LSMGSNnμ,Σ,β,Θ,π⋅ is given by(24)EϖZ1Z−μ=EθθΣE∇ϖS1+2θΣ1/2γEϖM1π′γTθΣ−1/2M−μ−θβ+EθEϖS1θβ,where π′⋅ is the derivative of π⋅, and(25)S=S1T,S2TT=ZΘ=θ∼GSNnμ+θβ,θΣ,π⋅,M=M1T,M2TT∼Nnμ+θβ,θΣ.Example 3.
Lettingπ⋅=Φ⋅(the cdf of a standard normal distribution) in Example 2, Stein-type lemma for location-scale mixture of skew-normal random vector is given by(26)EϖZ1Z−μ=EθθΣE∇ϖS1+2πθΣ1/2γEϖM1exp−12γTθΣ−1/2M−μ−θβ2+EθEϖS1θβ.Example 4.
Lettingπ⋅=1 in Example 2, Stein-type lemma for location-scale mixture of normal random vector is given by(27)EϖZ1Z−μ=EθθΣE∇ϖM1+EθEϖM1θβ.


## 5. Application in Risk Theory

Consideringn risky assets with stochastic returns that are modelled by the n-dimensional random vector,(28)Z=Z1,Z2,...,ZnT∼LSMGSEnμ,Σ,β,Θ,gn,π⋅.A risk-free asset bearing a fixed rate of returnr>0 is also available. Denote by t=t1,t2,...,tnT the vector of proportions that are allocated to the different risky assets. The total portfolio return is(29)Zt=∑i=1ntiZi+1−∑i=1ntiZir.We define(30)St=∑i=1ntiSi+1−∑i=1ntiSir,St∗=∑i=1ntiSi∗+1−∑i=1ntiSi∗r,Mt∗=∑i=1ntiMi∗+1−∑i=1ntiMi∗r.To find an optimal allocation by maximizing the mean return for a given variance risk tolerance, we assume that the investor optimizes(31)Lt:=EUZt,where U:ℝ⟶ℝ is a concave utility function (see [11]).Theorem 4.
(Three-Fund Separation). SupposeLt is a concave continuously differentiable function with t∈ℝn. The solution to problem (31) is given as(32)tLSMGSE=Σ−1δ1ψ′0EθθEU″StLSMGSE∗+δ2EθθEU″StLSMGSE∗,where e=1,1,...,1T is an n×1 vector whose elements are all equal to 1, and(33)δ1=EθEU′StLSMGSEθβ+EU′ZtLSMGSEμ−re,δ2=Eθ2θEU′MtLSMGSE∗∇πθΣ−1/2M∗−μ−θβ.Proof.
Letting∇Lt=0, we have(34)EU′ZtZ−re=0,EU′ZtZ−μ+EU′Ztμ−re=0.
Using (19), we get(35)EθCovME∇U′St∗+2EU′Mt∗∇πθΣ−1/2M∗−μ−θβ+EθEU′Stθβ+EUZtμ−re=0.
Note thatE∂U′Zt∗/∂Zi∗=EtiU″Zt∗ and CovM=−ψ′0θΣ; we have(36)Eθ−ψ′0θΣEtU″St∗+2EU′Mt∗∇πθΣ−1/2M∗−μ−θβ+EθEU′Stθβ+EU′Ztμ−re=0.
Therefore, we obtain (32), which completes the proof of Theorem 4.Remark 5.
WhenL⋅ is only defined on a convex subset A of ℝn, a solution t to (32) is only optimal when t belongs to the interior of this set A. Otherwise, the optimum has to be found on the boundary of A (see [12] or [11]).Corollary 1.
Lettingπ⋅=1 in Theorem 4, we obtain(37)tLSME=Σ−1δ1ψ′0EθθEU″MtLSME∗,where δ1=EθEU′MtLSMEθβ+EU′ZtLSMEμ−re.



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