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

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.

1. Introduction and Motivation

Since Stein [1] provides an expression for normal random variable , where 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 is said to be an -dimensional normal mean-variance mixture variable if , where , the -dimensional normal random vectors with the identity covariance matrix; is an matrix; is a scalar random variable that follows a nonnegative distribution with the density , independent of ; and the following are sontant vectors in :These specification implies that conditionally, , where . 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. Section 2 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.

Let be an -dimensional generalized skew-elliptical random vector and denoted by . If its probability density function exists, the form will be (see [6])whereis the density of -dimensional elliptical random vector . Here, is an location vector, is an scale matrix, and , , is the density generator of . , is called the skewing function satisfying and . The characteristic function of takes the form , with function , called the characteristic generator (see [9]). Suppose be an matrix and be an vector. Then,

To establish Stein’s lemma for -dimensional generalized skew-elliptical distributions, we use the cumulative generator . It takes the following form (see [7] or [10]):

Let be an elliptical random vector with generator , whose density function (if it exists) is

Let be a generalized skew-elliptical random vector.

We call as an -dimensional LSMGSE distribution with location parameter , positive definite scale matrix , and skew function , ifwhere and . Assume that is independent of nonnegative scalar random variable . We have

3. Main Result

In this section, we consider a random vectorwith location parameter , positive definite scale matrix , and skew function as (7).

Let , be an almost everywhere differentiable function, and we write

We derive a Stein-type lemma for location-scale mixture of generalized skew-elliptical random vectors below. Partition , where and . and are also of similar partition.

Theorem 1. Let be an -dimensional location-scale mixture of generalized skew-elliptical random vector defined as (7). Assume that function satisfies . Then,where

Proof. Using tower property of expectations, we obtainwhilewhere 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 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. Let be an -dimensional location-scale mixture of generalized skew-elliptical random vector withwhere . Assume the function satisfies . Then,where

Proof. Letting in Theorem 1, we directly obtain (16). This completes the proof of Theorem 2.

Remark 2. Letting in Theorem 2, we obtain a Stein-type lemma for location-scale mixture of elliptical random vectors:

Theorem 3. Let be an -dimensional location-scale mixture of generalized skew-elliptical random vector defined as (7). Assume the function satisfies . Then,

Proof. Letting in Theorem 1, we obtain (19). This completes the proof of Theorem 3.

Remark 3. Letting in Theorem 3, we obtain a Stein-type lemma for location-scale mixture of elliptical random vectors:

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. Letting in Theorem 1, Stein-type lemma for location-scale mixture of elliptical random vector is given by

Remark 4. We find that (21) can be regarded as a special analogue case of Vanduffel and Yao [11].

Example 2. Suppose is an -dimensional generalized skew-normal random vector with probability density function (pdf) as follows:, where and function . Letting andin Theorem 1. Assuming that function satisfies , Stein-type lemma for location-scale mixture of generalized skew-normal random vector is given bywhere is the derivative of , and

Example 3. Letting (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

Example 4. Letting in Example 2, Stein-type lemma for location-scale mixture of normal random vector is given by

5. Application in Risk Theory

Considering risky assets with stochastic returns that are modelled by the -dimensional random vector,

A risk-free asset bearing a fixed rate of return is also available. Denote by the vector of proportions that are allocated to the different risky assets. The total portfolio return is

We define

To find an optimal allocation by maximizing the mean return for a given variance risk tolerance, we assume that the investor optimizeswhere is a concave utility function (see [11]).

Theorem 4. (Three-Fund Separation). Suppose is a concave continuously differentiable function with . The solution to problem (31) is given aswhere is an vector whose elements are all equal to 1, and

Proof. Letting , we haveUsing (19), we getNote that and ; we haveTherefore, we obtain (32), which completes the proof of Theorem 4.

Remark 5. When is only defined on a convex subset of , a solution to (32) is only optimal when belongs to the interior of this set . Otherwise, the optimum has to be found on the boundary of (see [12] or [11]).

Corollary 1. Letting in Theorem 4, we obtainwhere .

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (Nos. 11571198 and 11701319).