Stein’s lemma for truncated generalized skew-elliptical random vectors

1.   Introduction and motivation

Stein ^[3] provide an expression $E[h(X)(X-\mu)]$ for normal random variable $X$, where $h(x)$ is an almost differentiable function. Then a number of scholars have generalized the formula. For examples, Landsman ^[4], Landsman and Ne$\check{s}$lehov$\acute{a}$ ^[5], Landsman et al. ^[6] derive Stein's lemma for multivariate elliptical distributions. Adcock and Shutes ^[7], Adcock ^[8], Adcock et al. ^[2] derive Stein's lemma for multivariate skew distributions. Liu ^[9] use Stein's lemma derive the Siegel's formula, and Li ^[10], Landsman et al. ^[11] apply this lemma to study risk measures.

Recently, Shushi ^[1] provide Stein's lemma for truncated elliptical random vectors, and inspired by this, we shall generalize Stein's lemma for truncated generalized skew-elliptical distributions. As applications, we consider the conditional tail expectation (CTE) allocation, the lower-orthant CTE at probability level $q$, the upper-orthant CTE at probability level $q$, the truncated version of Wang's premium, the multivariate tail conditional expectation and the multivariate tail covariance matrix measures of Stein's lemma for generalized skew-elliptical random vectors.

The rest of the paper is organized as follows. Section 2 reviews the definitions and properties of the generalized skew-elliptical distributions. In Section 3, We provide two Stein's lemmas, one is Stein's lemma for truncated generalized skew-elliptical random vectors, the other is a special form of Stein's lemma for truncated generalized skew-elliptical random vectors. Several measures as applications in risk theory are given in Section 4. Conclusions are summarized in Section 5.

2.   Generalized skew-elliptical distributions

Let ${\bf{Y}}$ be an $n$-dimensional generalized skew-elliptical random vector, and denoted by ${\bf{Y}}\sim GSE_{n}\left(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot)\right)$. If it's probability density function exists, the form will be (see Adcock et al. ^[2])

where

is the density of $n$-dimensional elliptical random vector ${\bf{X}}\sim E_{n}(\boldsymbol{\mu}, \boldsymbol{\Sigma}, g_{n})$. Here $\boldsymbol{\mu}$ is an $n\times1$ location vector, ${\bf{\Sigma}}$ is an $n\times n$ scale matrix, and $g_{n}(u)$, $u\geq0$, is the density generator of ${\bf{X}}$. $\pi(\boldsymbol{x}), \; \boldsymbol{x}\in\mathbb{R}^{n},$ is called the skewing function satisfying $\pi(\boldsymbol{-x}) = 1-\pi(\boldsymbol{x})$ and $0\leq\pi(\boldsymbol{x})\leq1$. The characteristic function of ${\bf{X}}$ takes the form $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) = \exp\left\{i\boldsymbol{t}^{T}\boldsymbol{\mu}\right\}\psi\left(\frac{1}{2}\boldsymbol{t}^{T}\boldsymbol{\Sigma}\boldsymbol{t}\right), \; \boldsymbol{t}\in \mathbb{R}^{n}$, with function $\psi(t):[0, \infty)\rightarrow\mathbb{R},$ called the characteristic generator (see Fang et al. ^[12]). We definite a cumulative generator $\overline{G}_{n}(u)$. It takes the form (see Landsman et al. ^[11], Part 4 or Landsman ^[13])

3.   Main results

In this section, consider a random vector ${\bf{Y}}\sim GSE_{n}\left(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot)\right)$ with finite vector $\boldsymbol{\mu} = (\mu_1, \cdots, \mu_n)^{T}$, positive defined matrix ${\bf {\Sigma}} = (\sigma_{ij})_{i, j = 1}^{n}$ and probability density function $f_{\boldsymbol{Y}}(\boldsymbol{y})$.

Let $\varpi:\mathbb{R}^{m}\rightarrow\mathbb{R}, \; 1\leq m\leq n,$ be an almost differentiable function, and we write

Let ${\bf{X}}^{\ast}\sim E_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; \overline{G}_{n})$ be an elliptical random vector with generator $\overline{G}_{n}(u)$, whose the density function (if it exists)

Let ${\bf{Y}}^{\ast}\sim GSE_{n}(\boldsymbol{\mu}, \; {\bf{\Sigma}}, \; \overline{G}_{n}, \; \pi(\cdot))$ be a generalized skew-elliptical random vector.

Let's derive Stein's lemma for truncated generalized skew-elliptical random vectors below. Firstly, we define a subset $\mathbb{D}\subseteq\mathbb{R}^{n}$, which is a subset of all possible outcomes of ${\bf{Y}}\in\mathbb{R}^{n}$, and

Then partition ${\bf{Y}} = ({\bf{Y_{(1)}}}^{T}, \; {\bf{Y_{(2)}}}^{T})^{T}$, where ${\bf{Y_{(1)}}} = (Y_{1}, \; Y_{2}, \cdots, Y_{m})^{T}$ and ${\bf{Y_{(2)}}} = (Y_{m+1}, \; Y_{m+2}, \cdots, Y_{n})^{T}$. Furthermore, ${\bf{X}} = ({\bf{X_{(1)}}}^{T}, \; {\bf{X_{(2)}}}^{T})^{T}$ and $\boldsymbol{\mu} = (\boldsymbol{\mu_{(1)}}^{T}, \; \boldsymbol{\mu_{(2)}}^{T})^{T}$ are similar partitions.

$\bf{Theorem\; 3.1.}$ Let ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; {\bf{\Sigma}}, \; g_{n}, \; \pi(\cdot))$ be an $n$-dimensional generalized skew-elliptical random vector with probability density function $(2.1)$. The function $\varpi$ satisfies $E^{\mathbb{D}}[\parallel\nabla \varpi({\bf{Y_{(1)}}}^{\ast})\parallel] < \infty$, $E\left[\parallel \varpi({\bf{Y_{(1)}}}^{\ast})\nabla\boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}}\parallel\right] < \infty$ and $E^{\mathbb{D}}\left[\parallel \varpi({\bf{X_{(1)}}}^{\ast})\nabla \pi\left({\bf{\Sigma}} ^{-\frac{1}{2}}({\bf{X}}^{\ast}-\boldsymbol{\mu})\right)\parallel\right] < \infty$, where $\parallel\cdot\parallel$ is the Euclidean norm on $\mathbb{R}^{n}$. Furthermore, we suppose

Then

where $\nabla\boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}} = \left(\frac{\partial \boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}}}{\partial Y_{1}^{\ast}}, \frac{\partial \boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}}}{\partial Y_{2}^{\ast}}, \cdots, \frac{\partial \boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}}}{\partial Y_{n}^{\ast}}\right)^{T}$, and $\boldsymbol{1}$ is the indicator function. In addition, $\nabla \pi\left({\bf{\Sigma}} ^{-\frac{1}{2}}({\bf{X}}^{\ast}-\boldsymbol{\mu})\right) = \frac{\mathrm{d}}{\mathrm{d}{\bf{X}}^{\ast}}\pi\left({\bf{\Sigma}} ^{-\frac{1}{2}}({\bf{X}}^{\ast}-\boldsymbol{\mu})\right)$ and

Proof. Using definition, we obtain

Setting $\boldsymbol{z} = {\bf{\Sigma}}^{-\frac{1}{2}}(\boldsymbol{y-\mu})$, we have

where $\Pr(\boldsymbol{Z}\in\mathbb{D}_{\boldsymbol{Z}}) = \Pr(\boldsymbol{Y}\in\mathbb{D})$, and in the third equality we have used $(3.2)$.

While

so that

therefore we obtain $(3.3)$, which completes the proof of Theorem $3.1$.

As a special case, Stein's lemma for $n$-dimensional truncated generalized skew-normal distribution is shown as follows.

$\bf{Corollary\; 3.1.}$ Let ${\bf{Y}}\sim GSN_{n}(\boldsymbol{\mu}, \; \Sigma, \; \pi(\cdot))$ be an $n$-dimensional generalized skew-normal random vector with probability density function

where $\boldsymbol{\gamma} = (\gamma_{1}, \; \gamma_{2}, \; \cdots, \; \gamma_{n})^{T}$ and function $\pi(\cdot): \mathbb{R}\rightarrow\mathbb{R}$. Then

where ${\bf{X}}\sim N_{n}(\boldsymbol{\mu}, \; {\bf{\Sigma}})$, and $\pi'(\cdot)$ is the derivative of $\pi(\cdot)$.

Proof. Let $g_{n}(u) = (2\pi)^{-\frac{n}{2}}\exp\{-u\}$ and $\pi\left({\bf{\Sigma}}^{-\frac{1}{2}}(\boldsymbol{y}-\boldsymbol{\mu})\right) = \pi\left(\boldsymbol{\gamma}^{T}{\bf{\Sigma}}^{-\frac{1}{2}}(\boldsymbol{y}-\boldsymbol{\mu})\right)$ in Theorem $3.1$. Due to $g_{n}(u) = \overline{G}_{n}(u) = (2\pi)^{-\frac{n}{2}}\exp\{-u\}$, so that we obtain $(3.4)$. This completes the proof of Corollary $3.1$.

$\bf{Remark\; 3.1.}$ Let $\pi(\cdot) = \Phi(\cdot)$(the cdf of a standard normal distribution) in Corollary $3.1$, we obtain Stein's lemma for $n$-dimensional truncated skew-normal distribution as follows.

$\bf{Remark\; 3.2.}$ In Loperfido ^[14], a trivariate distribution with the following pdf was introduced:

where $\phi(\cdot)$ and $\Phi(\cdot)$ are pdf and the cdf of the standard normal distribution, respectively. Moreover, $a$ is a nonnull real value. The conditional distribution of $Y_{1}$ given that $Y_{2} = y_{2}$ and $Y_{3} = y_{3}$ is skew-normal with pdf

where $c = ay_{2}y_{3}$ is a real value. Stein's lemma for this distribution can be obtained from Remark 3.1.

The following theorem gives a special form of Stein's lemma for truncated generalized skew-elliptical random vectors.

$\bf{Theorem\; 3.2.}$ Let ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; {\bf{\Sigma}}, \; g_{n}, \; \pi(\cdot))$ be an $n$-dimensional generalized skew-elliptical random vector with probability density function $(2.1)$. The function $\varpi$ satisfies $E^{\mathbb{D}}[\parallel\nabla \varpi({\bf{Y}}^{\ast})\parallel] < \infty$, $E\left[\parallel \varpi({\bf{Y}}^{\ast})\nabla\boldsymbol{1}_{\boldsymbol{Y}^{\ast}\in\mathbb{D}}\parallel\right] < \infty$ and $E^{\mathbb{D}}\left[\parallel \varpi({\bf{X}}^{\ast})\nabla \pi\left({\bf{\Sigma}} ^{-\frac{1}{2}}({\bf{X}}^{\ast}-\boldsymbol{\mu})\right)\parallel\right] < \infty$, where $\parallel\cdot\parallel$ is the Euclidean norm on $\mathbb{R}^{n}$. Furthermore, we suppose

Then

Proof. Let $\varpi(\boldsymbol{y_{(1)}}) = \varpi(\boldsymbol{y})$ in Theorem $3.1$, we obtain $(3.6)$, which completes the proof of Theorem $3.2$.

$\bf{Remark\; 3.3.}$ Let $\pi(\cdot) = \frac{1}{2}$ in Theorem $3.2$, we obtain

which is an equivalent form of Theorem 1 in Shushi ^[1].

4.   Applications in risk theory

Let $S_{{\bf{X}}}$ denote the agammaegate or portfolio risk $S_{{\bf{X}}} = X_{1}+X_{2}+\cdots+X_{n}$, the risk allocation for the conditional tail expectation (CTE) is a rule that decomposes the CTE of $S_{{\bf{X}}}$ to each $X_{i}$ such that $E[S_{{\bf{X}}}|S_{{\bf{X}}} > s_{p}] = \sum_{j = 1}^{n}\rho(X_{j}|\Omega),$ where $\rho(X_{j}|\Omega)$ stands for the allocated risk to the $j$th line, and $\Omega = \{X_{1}, \; X_{2}, \; X_{n}\}$ represents the whole portfolio. Since $\rho(X_{j}|\Omega) = E[X_{j}|S_{{\bf{X}}} > s_{p}]$, so that (see Kim et al. ^[15])

We now derive $E[S_{{\bf{Y}}}|S_{{\bf{Y}}} > s_{p}]$ for generalized skew-elliptical random vectors.

$\bf{Theorem\; 4.1.}$ Consider ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot))$ is an $n$-dimensional generalized skew-elliptical random vector. Then the CTE allocation for $S_{{\bf{Y}}}$ is given by

where $S_{{\bf{Y}}} = Y_{1}+Y_{2}+\cdots+Y_{n}$, and $\boldsymbol{e} = (1, \; 1, \; \cdots, \; 1)^{T}$ is an $n\times1$ vector whose elements are all equal to $1$.

Proof. Let $\varpi({\bf{Y}}) = 1$, and ${\bf{Y}}\in\mathbb{D}$ subject to $S_{{\bf{Y}}^{\ast}} > s_{p}$ in Theorem $3.2$, we can obtain

Since $E[S_{{\bf{Y}}}|S_{{\bf{Y}}} > s_{p}] = \boldsymbol{e}^{T}E[{\bf{Y}}|S_{{\bf{Y}}} > s_{p}]$, so we can get formula $(4.1)$. This completes the proof of Theorem $4.1$.

In addition, we introduce two CTE measure as follows (see Cousin and Bernardino ^[16] or Shushi ^[1]).

The lower-orthant CTE at probability level $q$

The upper-orthant CTE at probability level $q$

Here $F$ is distribution function of ${\bf{X}}$, and $\overline{F}$ is survival function of ${\bf{X}}$.

We now give lower-orthant CTE and upper-orthant CTE for generalized skew-elliptical random vectors.

$\bf{Theorem\; 4.2.}$ Consider ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot))$ is an $n$-dimensional generalized skew-elliptical random vector. Then

Proof. Let $\varpi({\bf{Y}}) = 1$, and ${\bf{Y}}\in\mathbb{D}$ subject to $F({\bf{Y}})\geq q$ in Theorem $3.2$, we directly obtain $(4.3)$. Let $\varpi({\bf{Y}}) = 1$, and ${\bf{Y}}\in\mathbb{D}$ submit to $\overline{F}({\bf{Y}})\leq 1-q$ in Theorem $3.2$, we get $(4.4)$. This completes the proof of Theorem $4.2$.

The truncated version of Wang's premium can be defined, as follows (see Shushi ^[1]):

with the tuned exponential tilting $\exp\left\{\boldsymbol{\lambda}^{T}{\bf{X}}\right\}$ and $\boldsymbol{\lambda} = (\lambda_{1}, \; \lambda_{2}, \cdots, \lambda_{n})^{T}\in\mathbb{R}^{n},$ $q\in(0, \; 1)$, where moment generating function is exists, i.e.,

We now derive $\pi_{q, \; \boldsymbol{\lambda}}({\bf{Y}})$ for generalized skew-elliptical random vectors.

$\bf{Theorem\; 4.3.}$ Consider ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot))$ is an $n$-dimensional generalized skew-elliptical random vector, and satisfying formula $(4.5)$. Then

where

Proof. Substituting $\varpi({\bf{Y}}) = \exp\left\{\boldsymbol{\lambda}^{T}{\bf{Y}}\right\}$ into Theorem $3.2$, we obtain

so that

Therefore we obtain $(4.6)$, which completes the proof of Theorem $4.3$.

Landsman et al. ^[11] defined the multivariate tail conditional expectation (MTCE) measure

where $VaR_{\boldsymbol{q}}({\bf{Y}}) = (VaR_{q_{1}}(Y_{1}), \; VaR_{q_{2}}(Y_{2}), \cdots, VaR_{q_{n}}(Y_{n}))^{T}$, $VaR_{q_{i}}(Y_{i}) = y_{q_{i}}$ is the value at risk of $Y_{i}$ under the $q_{i}$-th quantile, $q_{i}\in[0, \; 1), \; i = 1, \; 2, \cdots, n,$ and $\boldsymbol{q} = (q_{1}, \; q_{2}, \cdots, q_{n}).$ In addition, we define multivariate tail covariance matrix

The following two theorems give $MTCE_{\boldsymbol{q}}({\bf{Y}})$ and $MTCov_{\boldsymbol{q}}({\bf{Y}})$, respectively.

$\bf{Theorem\; 4.4.}$ Consider ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot))$ is an $n$-dimensional generalized skew-elliptical random vector. Then

Proof. Let $\varpi({\bf{Y}}) = 1$, and ${\bf{Y}}\in\mathbb{D}$ subject to ${\bf{Y}} > VaR_{\boldsymbol{q}}({\bf{Y}})$ in Theorem $3.2$, we can obtain $(4.7)$. This is completes proof of Theorem $4.4$.

$\bf{Theorem\; 4.5.}$ Consider ${\bf{Y}}\sim GSE_{n}(\boldsymbol{\mu}, \; \boldsymbol{\Sigma}, \; g_{n}, \; \pi(\cdot))$ is an $n$-dimensional generalized skew-elliptical random vector. Then

where

and $\boldsymbol{e}_{i} = (0, \cdots, 0, \; 1, \; 0, \cdots, 0)^{T}$ is an $n\times1$ vector whose elements are all equal to zero except $i$-th element, which is equal to $1$.

Proof. Let $\varpi({\bf{Y}}) = Y_{i}$, and ${\bf{Y}}\in\mathbb{D}$ subject to ${\bf{Y}} > VaR_{\boldsymbol{q}}({\bf{Y}})$ in Theorem $3.2$, we have

Multiplying $E[Y_{i}({\bf{Y}}-\boldsymbol{\mu})|{\bf{Y}} > VaR_{\boldsymbol{q}}({\bf{Y}})]$ by $\boldsymbol{e}_{j}^{T}$ from the left, we get

So that

where in the last line we have used the following relation

While

where $MCTE_{\boldsymbol{q}}({\bf{Y}}) = \boldsymbol{p} = (p_{1}, \; p_{2}, \cdots, p_{n})^{T}$, $p_{i} = \boldsymbol{e}_{i}^{T}MCTE_{\boldsymbol{q}}({\bf{Y}}), \; i = 1, \; 2, \cdots, n$, and in the fourth equality we have used $(4.10)$.

Using relations $(4.9)$ and $(4.11)$, we obtain $(4.8)$. This is completes proof of Theorem $4.5$.

5.   Conclusions

In this paper, we extended Stein's lemma in Shushi ^[1], deriving two Stein's lemmas for truncated generalized skew-elliptical random vectors. Moreover, as applications in risk theory, we obtained the expressions for the conditional tail expectation (CTE) allocation, the lower-orthant CTE at probability level $q$, the upper-orthant CTE at probability level $q$, the truncated version of Wang's premium, the multivariate tail conditional expectation and the multivariate tail covariance matrix measures. Our results also possibly apply to model of financial returns, for example, the multivariate SGARCH model proposed by De Luca, Genton and Loperfido ^[17] and further studied by De Luca and Loperfido ^[18]. We hope that these important problems can be addressed in future research.

Acknowledgments

The authors would like to thank two anomymous referees for their very useful comments which greatly helped in improving the quality of the present paper. The research was supported by the National Natural Science Foundation of China (No. 11171179, 11571198, 11701319).

Conflicts of interest

The authors declare that they have no conflicts of interest.