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Stein’s lemma for truncated elliptical random vectors
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Stein’s lemma for truncated elliptical random vectors Tomer Shushi Actuarial Research Center, Department of Statistics, University of Haifa, Israel Department of Economics and Business Administration, Ariel University, Israel
article
a b s t r a c t
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Article history: Received 13 September 2017 Received in revised form 29 January 2018 Accepted 9 February 2018 Available online xxxx
In this letter we derive the multivariate Stein’s lemma for truncated elliptical random vectors. The results in this letter generalize Stein’s lemma for elliptical random vectors given in Landsman and Nešlehová (2008), and the tail Stein’s lemma given in Landsman and Valdez (2016). We give a conditional Stein’s-type inequalities and a conditional version of Siegel’s formula for the elliptical distributions, and by that we generalize results obtained in Landsman et al. (2013) and in Landsman et al. (2015). Furthermore, we show applications of the main results in the letter for risk theory. © 2018 Elsevier B.V. All rights reserved.
Keywords: Capital asset pricing models Density generator Elliptical distributions Siegel’s formula Stein’s lemma Truncated random vectors
1. Introduction
1
Stein’s lemma (Stein, 1981) gives an important and elegant formula for the multivariate normal distributions, and has many applications in quantitative finance and statistics (Froot, 2007; Landsman and Nešlehová, 2008; Adcock, 2014; Gron et al., 2012; Vanduffel and Yao, 2017). In quantitative finance, this lemma is used to calculate capital asset pricing models for returns of arbitrary number of dependent assets (Fama and French, 2004; Levy, 2012; Barberis et al., 2015). Furthermore, a vast number of models deal with asset returns as truncated random variables. For example, value at risk measure, tail value at risk measure, truncated regression models, and censored quantile regressions, are models that are based on truncated random variables (Liu, 1994; Cousin and Di Bernardino, 2014; Landsman et al., 2016; Kong and Xia, 2017). Therefore, it seems natural to generalize Stein’s lemma for truncated random vectors. (X1⏐,)X2 )T be a bivariate normal random vector and consider a differentiable function h : R2 → R such that (⏐Let ′ ⏐ E h (X1 )⏐ < ∞. Then, the bivariate Stein’s lemma states that (see, for instance, Landsman and Nešlehová, 2008)
) Cov (h (X1 ) , X2 ) = Cov (X1 , X2 ) · E h (X1 ) . (
′
2 3 4 5 6 7 8 9 10 11
12
Suppose we have an n-variate normal random vector X ∽ Nn (µ, Σ ), and a differentiable function h : R → R such that the expectation of the norm of ∇ h(X) = (∂ h (x) /∂ x1 , ∂ h (x) /∂ x2 , . . . , ∂ h (x) /∂ xn )T exists. Then, Stein’s lemma is given by Stein (1981) n
Cov (h (X) , X) = Σ E (∇ h (X)) . In the following section we give a definition for the family of elliptical distributions and describe several of its important properties. In Section 3 we derive Stein’s lemma for truncated elliptical random vectors, we then generalize several results about Stein’s identity and Siegel’s formula, and show important inequalities of the proposed truncated Stein’s lemma. Section 4 presents applications of the main results in the letter for risk theory. E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2018.02.008 0167-7152/© 2018 Elsevier B.V. All rights reserved.
Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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2. The elliptical distributions The family of elliptical distributions is an extension of the normal distribution (Cambanis et al., 1981) into a broader family. Let X be n × 1 random vector following elliptical distribution, X ∽ En (µ, Σ , gn ). Then, the probability density function (pdf) of X is fX (x) = √
(
1
gn
1
)
(x − µ)T Σ −1 (x − µ) , x ∈ R,
(1)
2 |Σ | where gn (u) , u ≥ 0, is called the density generator of X, µ is an n × 1 vector of means, and Σ is an n × n scale matrix. The characteristic function of X takes the form ϕX (t) = exp(itT µ)ψ ( 12 tT Σ t), t ∈ Rn , with some function ψ (u) : [0, ∞) → R, σ2
called the characteristic generator. The covariance matrix of X is then given by Cov (X) = nZ Σ , where σZ2 = −ψ ′ (0). For the sequel, we define a cumulative generator function Gn (u) (see, for instance, Landsman et al. (2016)), such that ∞
∫
gn (x) dx,
Gn (u) =
10
(2)
u 11
and an associated elliptical random vector X∗ ∽ En (µ, Σ , nGn /σZ2 ) whose pdf takes the form fX∗ (t) =
12
13
14
16
18 19
20 21
26
27 28
(t − µ) Σ
−1
)
(t − µ) , t ∈ Rn .
(3)
E ∇ h X∗ < ∞,
(
(
))
(4)
where ∇ = d/dx is the n-multivariate operator of first derivatives, and ∥·∥ is the Euclidean norm on R . To present Stein’s lemma for truncated elliptical random vectors, we define a subset of Rn , R ⊆ Rn which is a subset of all possible outcomes of X( ∈ Rn , and) a conditional expected value E R (h (X) (X − µ)) := E (h (X) (X − µ) |X ∈ R) with the T conditional covariance of h (X) , XT , Cov R (h (X) , X) := E R (h (X) (X − µ)) − E (h (X) |X ∈ R) · E (X − µ|X ∈ R). m
Theorem 1. Let X be an elliptical random vector, X ∽ En (µ, Σ , gn ). Then, Stein’s lemma for the truncated random vector X| (X ∈ R) takes the form
) F ∗ (R)
E R (h (X) (X − µ)) = Cov (X) E ∇ h X∗ |X∗ ∈ R
(
Here E δ h X∗
( (
24
25
2
T
We define an almost differentiable function h : Rm → R, 1 ≤ m < n, under the following condition
22
23
Gn √ σZ2 |Σ |
1
3. Stein’s lemma for truncated elliptical random vectors
15
17
(
n
))
(
)
F (R)
( ( )) − E δ h X∗ .
(5)
( ( ) ( )) σZ = E h X∗ δ X∗ ∈ R √ , nF (R)
where
( ) ( δ X∗ ∈ R = δCov(X)11/2
δCov(X)1/2
...
2
δCov(X)1/2
)
(6)
n
1/2 i
is an n × 1 vector of surface delta functions δa = −a · ∇ 1X∗ ∈R (see, Lange (2012)), Cov(X) Cov(X)1/2 , F (R) = Pr (X ∈ R) , and F ∗ (R) = Pr (X∗ ∈ R) .
, i = 1, 2, . . . , n is the ith row of
Proof. Using the indicator function 1x∈R and under the linear transformation Σ −1/2 (X − µ) = z, we have E (h (X) (X − µ) |X ∈ R)
=
1
|Σ |1/2 F (R)
= F (R)−1 Σ 1/2
∫ ∫R Rn
) (X − µ)T Σ −1 (X − µ) dx 2 ) ( ( ) 1 T 1/2 z z dz, h µ + Σ z 1z∈RZ · zgn
h(x) (X − µ) gn
(
1
2
where the set RZ is such that {X ∈ R} = {Z ∈ RZ }. Taking into account the density generator G (u) (2) and the associated pdf (3), similar to Proposition 2 in Landsman et al. (2015) after partial integration and algebraic calculations, we observe that ( ( ) ) F ∗ (R) E (h (X) (X − µ) |X ∈ R) = Cov (X) E ∇ h X∗ |X∗ ∈ R F (R)
+ √
σZ
nF (R)
Cov(X)1/2 E h X∗ ∇ 1X∗ ∈R .
( (
)
)
Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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Finally, using the surface delta function, we conclude that E R (h (X) (X − µ)) = Cov (X) E ∇ h X
(
(
) ∗
1
) F ∗ (R)
|X∗ ∈ R
F (R)
−E
( ( δ
h X
)) ∗
.
■
Remark 1. To calculate the truncated Stein’s lemma for different members of the elliptical family we should find the pdf of X∗ . For instance, in the case that X ∽ Nn (µ, Σ ) the density generator is gn (u) = (2π)−n/2 e−u and thus Gn (u) = (2π)−n/2 e−u ,
2
3 4
d
which means that X = X∗ ∽ Nn (µ, Σ ) (see, for instance, Landsman et al. (2016)). For other important examples, see Landsman and Nešlehová (2008). Theorem 2. Let X ∽ Nn (µ, Σ ). Then, the truncated Stein’s lemma for the multivariate normal distribution is given by E
R
δ
(h (X) (X − µ)) = Σ E (∇ h (X) |X ∈ R) − E (h (X)) .
5 6
7 8
d
Proof. Using Eq. (5), by noticing that X = X∗ ∽ Nn (µ, Σ ) and Cov (X) = Σ , the theorem is then immediately proved. ■ We now show Stein’s identity in the case of truncated elliptical random variables.
9
10
Corollary 1. Let X ∽ En (µ, Σ , gn ), π ∈ Rn be a vector of constants. We describe the portfolio asset return as the weighted-sum R = π T X, where π1 , π2 , . . . , πn represent the weights of the asset returns X. Then, Stein’s identity for R| (R ∈ R) is given by E R h (R) R − π T µ
(
(
))
( ( ) ) F ∗ (R) = π T Σ πσ 2Z E h′ R∗ |R∗ ∈ R F (R) √ ( ( ∗) ( ∗ )) σZ2 , − πT ΣπE h R δ R ∈ R F (R)
where R∗ ∽ E1 (π T µ, π T Σ π, G1 /σZ2 ), and δ is the Dirac delta measure. Proof. From the marginality property of the elliptical distributions (see, for instance, Cambanis et al. (1981)), we conclude that R = π T X ∽ E1 (π T µ, π T Σ π, g1 ). Then, the proof immediately follows from Theorem 1, by taking the univariate random variable R and the δ -function instead of the surface δ -function. ■ Recently, Landsman et al. (2013) generalized the important Siegel’s formula (see, Siegel (1993)) for elliptical random vectors. We now show a generalization of Siegel’s formula for truncated elliptical random vectors.
11
12 13 14 15 16
Theorem 3. Let ( X ∽ En (µ, Σ ) , gn ), and let X(i) be the ith largest among the variables X1 , X2 , . . . , Xn . Then, the conditional covariance Cov X(i) , X1 |X ∈ R can be expressed, as follows: Cov X(i) , X1 |X ∈ R =
(
)
n ∑
Cov X1 , Xj
(
j=1
(( ) ) ) Pr Xj = X(i) ∩ (X∗ ∈ R) F (R)
( ∗) − E δ X(i) − E (Xi |X ∈ R) · E (X1 − µ1 |X ∈ R) , ∗ where X(i) is the ith largest among the variables X1∗ , X2∗ , . . . , Xn∗ .
Proof. Similar to Liu (1994) and Landsman et al. (2013), we define h as the piecewise linear function h (x) = x(i) and thus it is almost everywhere differentiable where
∂ h (x) /∂ xj =
{
1, xj = x(i) 0, xj ̸ = x(i) ,
17
18 19
20
substitute it in (5), and noticing that
21
( ( ) ) (( ) ( )) ( ) E ∇i h X∗ |X∗ ∈ R = Pr Xj = X(i) ∩ X∗ ∈ R /F R∗ , after some algebraic calculations the theorem is proved. ■ Theorem 4. Let X ∽ En (µ, Σ , gn ) where the scale matrix Σ has non-negative entries and δ (X∗ ∈ R) has non-positive components. Then, for a non-negative function h h : R → R+ ∪ {0}, 1 ≤ m < n, m
23
24 25 26
the following vector inequality holds
) F ∗ (R)
Cov R (h (X) , X) ≥ Cov (X) E ∇ h X∗ |X∗ ∈ R
(
22
(
)
F (R) − E (h (X) |X ∈ R) · E (X − µ|X ∈ R) .
Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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Proof. Since Σ has non-negative entries, and h is a non-negative function, we observe that E δ (h (X∗ )) ≤ 0. By omitting this term in (5), the theorem is immediately proved. ■
4
We now generalize a result obtained in Landsman et al. (2015) about Stein’s-type inequality for the generalized hyperbolic (GH) distributions.
5
Theorem 5. Let X ∽ GHn (µ, Σ , χ, ψ ) be a GH random vector with the pdf (Landsman et al., 2015)
3
6
(√ ( (√ )−λ n/2 )) χ ψ ψ Kλ−n/2 ψ χ + (x − µ)T Σ −1 (X − µ) , f (x) = ( ) (√ ) √ ( ) n/2−λ √ T n/2 − 1 |Σ |Kλ χψ ψ χ + (X − µ) Σ (X − µ) (2π) where Kλ is the modified Bessel function of the second kind with index λ and the different parameters satisfy χ > 0, ψ ≥ 0 if λ < 0, χ > 0, ψ > 0 if λ = 0, and χ ≥ 0, ψ > 0 if λ > 0. Let h : Rn → R be differentiable and non-increasing function for every component, and let Σ be a matrix with non-negative entries. Then, the following inequality holds
(√ ) χ 1/2 Kλ+1−n/2 χψ (√ ) Σ E (∇ h (X) |X ∈ R) ψ 1/2 Kλ−n/2 χψ ( ( )) − E δ h X∗ − E (h (X) |X ∈ R) · E (X − µ|X ∈ R) .
Cov R (h (X) , X) ≤
7 8 9
10
(7)
Proof. The proof of (7) relies heavily on techniques that were used in the proof of Theorem 1 in Landsman et al. (2015). Since the GH distribution is a special case of the family of elliptical distributions (see, again, Landsman et al. (2015)), Theorem 1 holds . Focusing on E (∇i h (X∗ ) |X∗ ∈ R) , we notice that E ∇i h X∗ |X∗ ∈ R =
(
(
)
)
∫
∇i h (x) fX (x) Cλ (x) dx/F ∗ (R) , R
11
12
13
14
where Cλ (x) =
fX∗ (x) fX (x)
.
In Landsman et al. (2015) it was shown that Cλ can be written in the following way
(√ ) χψ Kυ+1 (t ) · t (√ ) , χ ψ Kυ+1 χψ Kυ (t )
Kλ
˜ Cλ (t ) = √
where t = (X −√µ)T Σ −1 (x − µ) , υ = λ − n/2. Consider the case in which tKυ+1 (t ) /Kυ (t ) is a non-decreasing function. Then, since t ≥ ψχ , and ˜ Cλ (t ) is non-decreasing at t for fixed λ and ∇i h (x) ≤ 0, we have E ∇i h X∗ |X∗ ∈ R
(
(
)
)
∫
(√ ) ψχ dx/F ∗ (R) ∇i h (x) fX (x) ˜ Cλ ∗ R (√ ) (√ ) √ χ ψ Kυ+1 ψχ ψχ Kλ F (R) (√ ) (√ ) E (∇i h (X) |X ∈ R) ∗ = √ . F (R) χ ψ Kυ+1 χψ Kυ ψχ ≤
15
Finally, substituting (8) in (5) and after some algebraic calculations, the theorem is then proved. ■
16
4. Applications in risk theory
17 18 19 20 21 22 23 24 25 26 27
(8)
Recently, there is a growing interest in multivariate conditional risk measures that take into account the covariance structure of dependent risks. The importance of such measures comes from the fact that risks are oftenly depending on each other and unlike univariate risk measures, the multivariate risk measures consider the dependence structure of the risks which becomes important when the correlations between the risks are not weak (Jouini et al., 2004; Molchanov and Cascos, 2016; Feinstein and Rudloff, 2017; Landsman et al., 2016). Furthermore, a vast number of papers introduced and investigated models of risks for the elliptical family of distributions (see, for instance, Owen and Rabinovitch (1983), Valdez and Chernih (2003), Xiao and Valdez (2015), Landsman et al. (2016)). A well-known univariate risk measure is the conditional tail expectation (CTE), CTEq (Y ) := E (Y |F (Y ) ≥ q) , for some risk Y with distribution function F , and some probability level q ∈ (0, 1). In Cousin and Di Bernardino (2014) the authors introduced the following two multivariate versions of the CTE measure: Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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• The lower-orthant CTE at probability level q
1
MCTE q (X) = E (X|F (X) ≥ q) , q ∈ (0, 1).
(9)
• The upper-orthant CTE at probability level q ( ) MCTE q (X) = E X|F (X) ≤ 1 − q , q ∈ (0, 1).
3
(10)
These measures calculate the expected value of a random vector of risks X subject to a condition about its distribution function. The lower-orthant CTE (9) measures the conditional expected value of X subject to the condition that its distribution function, F , is greater than or equal to some probability level q, which directly implies on the case of extreme events of X. The upper-orthant CTE is interpreted in the same way as the lower-orthant measure, only in this case the condition is about the survival function of X, F , which assumes that the survival function of X is below or equal to some probability threshold level 1 − q. Both MCTE q (X) and MCTE q (X) compute that expectation of X around its tail which implies an extreme loss of the risks. Typical values for q are 0.9, 0.95, and 0.99 (see, for example, Shushi (2017)). Another extension of the CTE measure into the multivariate risk measures framework can be found in Landsman et al. (2016), which also derived such extended measure for the elliptical family of distributions. Wang’s premium principle takes the form (Wang, 2002) E Xi e λS
(
πλ (Xi , S) =
E e λS
(
7 8 9 10 11 12 13
15
16 17 18 19
)
λT X
|F (X) ≥ q ) , λi ≥ 0, T E eλ X |F (X) ≥ q
E Xi e
5 6
14
This measure computes the expected value of the risks Xi , i = 1, 2, . . . , n, with a tuned exponential tilting eλS , where S = X1 + X2 + · · · + Xn is the sum of the risks with finite moment generating function (mgf). This measure was computed for the elliptical distributions in Valdez and Chernih (2003), other premium principles for the elliptical distributions can be found in Landsman (2004). The truncated version of Wang’s premium can be defined, as follows:
πq,λ (Xi , S) =
4
)
) , λ ≥ 0.
(
2
20
(
with the tuned exponential tilting eλ X , λ = (λ1 , λ2 , . . . , λn )T ∈ Rn , under the assumption that the distribution function of the risks is greater than or equal to some threshold probability level q ∈ (0, 1), where the mgf exists, i.e., T
(
λT X
MX (λ) := E e
)
< ∞.
(11)
We now derive the truncated Wang’s measure for elliptical random vectors.
πq,λ (X) = µ + Cov (X) λαq,λ − Cov(X)1/2 E δ e
λT X∗
)
βq,λ .
23
25
26
Here αq,λ and βq,λ take the following explicit forms, respectively, MX∗ |(F (X∗ )≥q) (λ) Pr (F (X ) ≥ q)
22
24
Theorem 6. Let X ∽ En (µ, Σ , gn ). Under condition (11), the truncated Wang’s premium measure of X takes the following form
(
21
27
∗
αq,λ =
MX|(F (X)≥q) (λ) Pr (F (X) ≥ q)
,
28
and
29
βq,λ = MX|(F (X)≥q) (λ) , ( T ) where MX|(F (X)≥q) (λ) := E eλ X |F (X) ≥ q is the mgf of X| (F (X) ≥ q) . −1
Proof. Substituting h (X) = eλ
(
E e
λT X
TX
30
31
in Eq. (5), we have
( ) F ∗ (R) ( ) T ∗ (X − µ) |F (X) ≥ q = Cov (X) E ∇ eλ X |F X∗ ≥ q F (R) ( T ∗) 1/2 δ λ X − Cov(X) E e . )
Noticing that F (R) = Pr (F (X) ≥ q), and after some algebraic calculations, we conclude that
(
E eλ
TX
)
X|F (X) ≥ q = Cov (X) λ
MX∗ |(F (X∗ )≥q) (λ) F ∗ (R) F (R)
(
− Cov(X)1/2 E δ eλ
) ∗
TX
(12)
+ µMX∗ |(F (X∗ )≥q) (λ) .
Finally, dividing (12) by MX∗ |(F (X∗ )≥q) (λ), the theorem is then proved. ■ Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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We now derive explicit formulas for the upper and lower orthant MCTE risk measures for the family of elliptical distributions.
3
Theorem 7. The lower and upper orthant MCTE risk measures of X ∽ En (µ, Σ , gn ) take the following forms
1
σZ
MCTE q (X) = µ + √ Cov(X)1/2 E ∇ 1(F (X∗ )≥q) . n Pr (F (X) ≥ q)
4
5
6
7
and
(
σZ
)
(
)
MCTE q (X) = µ + √ ( ) Cov(X)1/2 E ∇ 1(F (X∗ )≤1−q) , n Pr F (X) ≤ 1 − q respectively. Proof. For the lower orthant (9), taking into account Theorem 6, and considering λ = 0, we observe that MX∗ |(F (X)≥q) (0) = 1, F (R) = Pr (F (X) ≥ q) , and βq,0 = 1, so
( ) σZ Cov(X)1/2 E ∇ 1(F (X∗ )≥q) βq,0 nF (R) ( ) Cov(X)1/2 E ∇ 1(F (X∗ )≥q) .
MCTE q (X) = µ + Cov (X) λαq,λ + √
σZ n Pr (F (X) ≥ q)
=µ+ √
For the upper orthant (10), we have E ((X − µ) |F (X) ≤ 1 − q)
= √
σZ
( T ∗ ) ) Cov(X)1/2 E eλ X ∇ 1(F (X∗ )≤1−q) ,
n Pr F (X) ≤ 1 − q
(
and after some algebraic calculations, we finally conclude that MCTE q (X) = E (X|F (X) ≤ 1 − q)
=µ+ √
σZ
( ) ) Cov(X)1/2 E ∇ 1(F (X∗ )≤1−q) .
n Pr F (X) ≤ 1 − q
8
9
10
(
■
Uncited references Landsman and Valdez (2016) Acknowledgments
12
I would like to thank the anonymous referee for the useful comments. This research was supported by the Israel Science Foundation (Grant No. 1686/17).
13
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Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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Molchanov, I., Cascos, I., 2016. Multivariate risk measures: a constructive approach based on selections. Math. Finance 26, 867–900. Owen, J., Rabinovitch, R., 1983. On the class of elliptical distributions and their applications to the theory of portfolio choice. J. Finance 38, 745–752. Shushi, T., 2017. Skew-elliptical distributions with applications in risk theory. Eur. Actuar. J. 7, 1–20. Siegel, A.F., 1993. A surprising covariance involving the minimum of multivariate normal variables. J. Amer. Statist. Assoc. 88, 77–80. Stein, C.M., 1981. Estimation of the mean of a multivariate normal distribution. Ann. Statist. 113, 5–1151. Valdez, E.A., Chernih, A., 2003. Wang’s capital allocation formula for elliptically contoured distributions. Insurance Math. Econom. 33, 517–532. Vanduffel, S., Yao, J., 2017. A stein type lemma for the multivariate generalized hyperbolic distribution. European J. Oper. Res. 261, 606–612. Wang, S., 2002. A Set of New Methods and Tools for Enterprise Risk Capital Management and Portfolio Optimization. Working Paper. SCOR Reinsurance Company. Xiao, Y., Valdez, E.A., 2015. A Black–Litterman asset allocation model under Elliptical distributions. Quant. Finance 15, 509–519.
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1 2 3 4 5 6 7 8 9 10 11
Model 3G
pp. 1–7 (col. fig: nil)
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Stein’s lemma for truncated elliptical random vectors Tomer Shushi Actuarial Research Center, Department of Statistics, University of Haifa, Israel Department of Economics and Business Administration, Ariel University, Israel
article
a b s t r a c t
info
Article history: Received 13 September 2017 Received in revised form 29 January 2018 Accepted 9 February 2018 Available online xxxx
In this letter we derive the multivariate Stein’s lemma for truncated elliptical random vectors. The results in this letter generalize Stein’s lemma for elliptical random vectors given in Landsman and Nešlehová (2008), and the tail Stein’s lemma given in Landsman and Valdez (2016). We give a conditional Stein’s-type inequalities and a conditional version of Siegel’s formula for the elliptical distributions, and by that we generalize results obtained in Landsman et al. (2013) and in Landsman et al. (2015). Furthermore, we show applications of the main results in the letter for risk theory. © 2018 Elsevier B.V. All rights reserved.
Keywords: Capital asset pricing models Density generator Elliptical distributions Siegel’s formula Stein’s lemma Truncated random vectors
1. Introduction
1
Stein’s lemma (Stein, 1981) gives an important and elegant formula for the multivariate normal distributions, and has many applications in quantitative finance and statistics (Froot, 2007; Landsman and Nešlehová, 2008; Adcock, 2014; Gron et al., 2012; Vanduffel and Yao, 2017). In quantitative finance, this lemma is used to calculate capital asset pricing models for returns of arbitrary number of dependent assets (Fama and French, 2004; Levy, 2012; Barberis et al., 2015). Furthermore, a vast number of models deal with asset returns as truncated random variables. For example, value at risk measure, tail value at risk measure, truncated regression models, and censored quantile regressions, are models that are based on truncated random variables (Liu, 1994; Cousin and Di Bernardino, 2014; Landsman et al., 2016; Kong and Xia, 2017). Therefore, it seems natural to generalize Stein’s lemma for truncated random vectors. (X1⏐,)X2 )T be a bivariate normal random vector and consider a differentiable function h : R2 → R such that (⏐Let ′ ⏐ E h (X1 )⏐ < ∞. Then, the bivariate Stein’s lemma states that (see, for instance, Landsman and Nešlehová, 2008)
) Cov (h (X1 ) , X2 ) = Cov (X1 , X2 ) · E h (X1 ) . (
′
2 3 4 5 6 7 8 9 10 11
12
Suppose we have an n-variate normal random vector X ∽ Nn (µ, Σ ), and a differentiable function h : R → R such that the expectation of the norm of ∇ h(X) = (∂ h (x) /∂ x1 , ∂ h (x) /∂ x2 , . . . , ∂ h (x) /∂ xn )T exists. Then, Stein’s lemma is given by Stein (1981) n
Cov (h (X) , X) = Σ E (∇ h (X)) . In the following section we give a definition for the family of elliptical distributions and describe several of its important properties. In Section 3 we derive Stein’s lemma for truncated elliptical random vectors, we then generalize several results about Stein’s identity and Siegel’s formula, and show important inequalities of the proposed truncated Stein’s lemma. Section 4 presents applications of the main results in the letter for risk theory. E-mail address: [email protected]. https://doi.org/10.1016/j.spl.2018.02.008 0167-7152/© 2018 Elsevier B.V. All rights reserved.
Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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2. The elliptical distributions The family of elliptical distributions is an extension of the normal distribution (Cambanis et al., 1981) into a broader family. Let X be n × 1 random vector following elliptical distribution, X ∽ En (µ, Σ , gn ). Then, the probability density function (pdf) of X is fX (x) = √
(
1
gn
1
)
(x − µ)T Σ −1 (x − µ) , x ∈ R,
(1)
2 |Σ | where gn (u) , u ≥ 0, is called the density generator of X, µ is an n × 1 vector of means, and Σ is an n × n scale matrix. The characteristic function of X takes the form ϕX (t) = exp(itT µ)ψ ( 12 tT Σ t), t ∈ Rn , with some function ψ (u) : [0, ∞) → R, σ2
called the characteristic generator. The covariance matrix of X is then given by Cov (X) = nZ Σ , where σZ2 = −ψ ′ (0). For the sequel, we define a cumulative generator function Gn (u) (see, for instance, Landsman et al. (2016)), such that ∞
∫
gn (x) dx,
Gn (u) =
10
(2)
u 11
and an associated elliptical random vector X∗ ∽ En (µ, Σ , nGn /σZ2 ) whose pdf takes the form fX∗ (t) =
12
13
14
16
18 19
20 21
26
27 28
(t − µ) Σ
−1
)
(t − µ) , t ∈ Rn .
(3)
E ∇ h X∗ < ∞,
(
(
))
(4)
where ∇ = d/dx is the n-multivariate operator of first derivatives, and ∥·∥ is the Euclidean norm on R . To present Stein’s lemma for truncated elliptical random vectors, we define a subset of Rn , R ⊆ Rn which is a subset of all possible outcomes of X( ∈ Rn , and) a conditional expected value E R (h (X) (X − µ)) := E (h (X) (X − µ) |X ∈ R) with the T conditional covariance of h (X) , XT , Cov R (h (X) , X) := E R (h (X) (X − µ)) − E (h (X) |X ∈ R) · E (X − µ|X ∈ R). m
Theorem 1. Let X be an elliptical random vector, X ∽ En (µ, Σ , gn ). Then, Stein’s lemma for the truncated random vector X| (X ∈ R) takes the form
) F ∗ (R)
E R (h (X) (X − µ)) = Cov (X) E ∇ h X∗ |X∗ ∈ R
(
Here E δ h X∗
( (
24
25
2
T
We define an almost differentiable function h : Rm → R, 1 ≤ m < n, under the following condition
22
23
Gn √ σZ2 |Σ |
1
3. Stein’s lemma for truncated elliptical random vectors
15
17
(
n
))
(
)
F (R)
( ( )) − E δ h X∗ .
(5)
( ( ) ( )) σZ = E h X∗ δ X∗ ∈ R √ , nF (R)
where
( ) ( δ X∗ ∈ R = δCov(X)11/2
δCov(X)1/2
...
2
δCov(X)1/2
)
(6)
n
1/2 i
is an n × 1 vector of surface delta functions δa = −a · ∇ 1X∗ ∈R (see, Lange (2012)), Cov(X) Cov(X)1/2 , F (R) = Pr (X ∈ R) , and F ∗ (R) = Pr (X∗ ∈ R) .
, i = 1, 2, . . . , n is the ith row of
Proof. Using the indicator function 1x∈R and under the linear transformation Σ −1/2 (X − µ) = z, we have E (h (X) (X − µ) |X ∈ R)
=
1
|Σ |1/2 F (R)
= F (R)−1 Σ 1/2
∫ ∫R Rn
) (X − µ)T Σ −1 (X − µ) dx 2 ) ( ( ) 1 T 1/2 z z dz, h µ + Σ z 1z∈RZ · zgn
h(x) (X − µ) gn
(
1
2
where the set RZ is such that {X ∈ R} = {Z ∈ RZ }. Taking into account the density generator G (u) (2) and the associated pdf (3), similar to Proposition 2 in Landsman et al. (2015) after partial integration and algebraic calculations, we observe that ( ( ) ) F ∗ (R) E (h (X) (X − µ) |X ∈ R) = Cov (X) E ∇ h X∗ |X∗ ∈ R F (R)
+ √
σZ
nF (R)
Cov(X)1/2 E h X∗ ∇ 1X∗ ∈R .
( (
)
)
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Finally, using the surface delta function, we conclude that E R (h (X) (X − µ)) = Cov (X) E ∇ h X
(
(
) ∗
1
) F ∗ (R)
|X∗ ∈ R
F (R)
−E
( ( δ
h X
)) ∗
.
■
Remark 1. To calculate the truncated Stein’s lemma for different members of the elliptical family we should find the pdf of X∗ . For instance, in the case that X ∽ Nn (µ, Σ ) the density generator is gn (u) = (2π)−n/2 e−u and thus Gn (u) = (2π)−n/2 e−u ,
2
3 4
d
which means that X = X∗ ∽ Nn (µ, Σ ) (see, for instance, Landsman et al. (2016)). For other important examples, see Landsman and Nešlehová (2008). Theorem 2. Let X ∽ Nn (µ, Σ ). Then, the truncated Stein’s lemma for the multivariate normal distribution is given by E
R
δ
(h (X) (X − µ)) = Σ E (∇ h (X) |X ∈ R) − E (h (X)) .
5 6
7 8
d
Proof. Using Eq. (5), by noticing that X = X∗ ∽ Nn (µ, Σ ) and Cov (X) = Σ , the theorem is then immediately proved. ■ We now show Stein’s identity in the case of truncated elliptical random variables.
9
10
Corollary 1. Let X ∽ En (µ, Σ , gn ), π ∈ Rn be a vector of constants. We describe the portfolio asset return as the weighted-sum R = π T X, where π1 , π2 , . . . , πn represent the weights of the asset returns X. Then, Stein’s identity for R| (R ∈ R) is given by E R h (R) R − π T µ
(
(
))
( ( ) ) F ∗ (R) = π T Σ πσ 2Z E h′ R∗ |R∗ ∈ R F (R) √ ( ( ∗) ( ∗ )) σZ2 , − πT ΣπE h R δ R ∈ R F (R)
where R∗ ∽ E1 (π T µ, π T Σ π, G1 /σZ2 ), and δ is the Dirac delta measure. Proof. From the marginality property of the elliptical distributions (see, for instance, Cambanis et al. (1981)), we conclude that R = π T X ∽ E1 (π T µ, π T Σ π, g1 ). Then, the proof immediately follows from Theorem 1, by taking the univariate random variable R and the δ -function instead of the surface δ -function. ■ Recently, Landsman et al. (2013) generalized the important Siegel’s formula (see, Siegel (1993)) for elliptical random vectors. We now show a generalization of Siegel’s formula for truncated elliptical random vectors.
11
12 13 14 15 16
Theorem 3. Let ( X ∽ En (µ, Σ ) , gn ), and let X(i) be the ith largest among the variables X1 , X2 , . . . , Xn . Then, the conditional covariance Cov X(i) , X1 |X ∈ R can be expressed, as follows: Cov X(i) , X1 |X ∈ R =
(
)
n ∑
Cov X1 , Xj
(
j=1
(( ) ) ) Pr Xj = X(i) ∩ (X∗ ∈ R) F (R)
( ∗) − E δ X(i) − E (Xi |X ∈ R) · E (X1 − µ1 |X ∈ R) , ∗ where X(i) is the ith largest among the variables X1∗ , X2∗ , . . . , Xn∗ .
Proof. Similar to Liu (1994) and Landsman et al. (2013), we define h as the piecewise linear function h (x) = x(i) and thus it is almost everywhere differentiable where
∂ h (x) /∂ xj =
{
1, xj = x(i) 0, xj ̸ = x(i) ,
17
18 19
20
substitute it in (5), and noticing that
21
( ( ) ) (( ) ( )) ( ) E ∇i h X∗ |X∗ ∈ R = Pr Xj = X(i) ∩ X∗ ∈ R /F R∗ , after some algebraic calculations the theorem is proved. ■ Theorem 4. Let X ∽ En (µ, Σ , gn ) where the scale matrix Σ has non-negative entries and δ (X∗ ∈ R) has non-positive components. Then, for a non-negative function h h : R → R+ ∪ {0}, 1 ≤ m < n, m
23
24 25 26
the following vector inequality holds
) F ∗ (R)
Cov R (h (X) , X) ≥ Cov (X) E ∇ h X∗ |X∗ ∈ R
(
22
(
)
F (R) − E (h (X) |X ∈ R) · E (X − µ|X ∈ R) .
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Proof. Since Σ has non-negative entries, and h is a non-negative function, we observe that E δ (h (X∗ )) ≤ 0. By omitting this term in (5), the theorem is immediately proved. ■
4
We now generalize a result obtained in Landsman et al. (2015) about Stein’s-type inequality for the generalized hyperbolic (GH) distributions.
5
Theorem 5. Let X ∽ GHn (µ, Σ , χ, ψ ) be a GH random vector with the pdf (Landsman et al., 2015)
3
6
(√ ( (√ )−λ n/2 )) χ ψ ψ Kλ−n/2 ψ χ + (x − µ)T Σ −1 (X − µ) , f (x) = ( ) (√ ) √ ( ) n/2−λ √ T n/2 − 1 |Σ |Kλ χψ ψ χ + (X − µ) Σ (X − µ) (2π) where Kλ is the modified Bessel function of the second kind with index λ and the different parameters satisfy χ > 0, ψ ≥ 0 if λ < 0, χ > 0, ψ > 0 if λ = 0, and χ ≥ 0, ψ > 0 if λ > 0. Let h : Rn → R be differentiable and non-increasing function for every component, and let Σ be a matrix with non-negative entries. Then, the following inequality holds
(√ ) χ 1/2 Kλ+1−n/2 χψ (√ ) Σ E (∇ h (X) |X ∈ R) ψ 1/2 Kλ−n/2 χψ ( ( )) − E δ h X∗ − E (h (X) |X ∈ R) · E (X − µ|X ∈ R) .
Cov R (h (X) , X) ≤
7 8 9
10
(7)
Proof. The proof of (7) relies heavily on techniques that were used in the proof of Theorem 1 in Landsman et al. (2015). Since the GH distribution is a special case of the family of elliptical distributions (see, again, Landsman et al. (2015)), Theorem 1 holds . Focusing on E (∇i h (X∗ ) |X∗ ∈ R) , we notice that E ∇i h X∗ |X∗ ∈ R =
(
(
)
)
∫
∇i h (x) fX (x) Cλ (x) dx/F ∗ (R) , R
11
12
13
14
where Cλ (x) =
fX∗ (x) fX (x)
.
In Landsman et al. (2015) it was shown that Cλ can be written in the following way
(√ ) χψ Kυ+1 (t ) · t (√ ) , χ ψ Kυ+1 χψ Kυ (t )
Kλ
˜ Cλ (t ) = √
where t = (X −√µ)T Σ −1 (x − µ) , υ = λ − n/2. Consider the case in which tKυ+1 (t ) /Kυ (t ) is a non-decreasing function. Then, since t ≥ ψχ , and ˜ Cλ (t ) is non-decreasing at t for fixed λ and ∇i h (x) ≤ 0, we have E ∇i h X∗ |X∗ ∈ R
(
(
)
)
∫
(√ ) ψχ dx/F ∗ (R) ∇i h (x) fX (x) ˜ Cλ ∗ R (√ ) (√ ) √ χ ψ Kυ+1 ψχ ψχ Kλ F (R) (√ ) (√ ) E (∇i h (X) |X ∈ R) ∗ = √ . F (R) χ ψ Kυ+1 χψ Kυ ψχ ≤
15
Finally, substituting (8) in (5) and after some algebraic calculations, the theorem is then proved. ■
16
4. Applications in risk theory
17 18 19 20 21 22 23 24 25 26 27
(8)
Recently, there is a growing interest in multivariate conditional risk measures that take into account the covariance structure of dependent risks. The importance of such measures comes from the fact that risks are oftenly depending on each other and unlike univariate risk measures, the multivariate risk measures consider the dependence structure of the risks which becomes important when the correlations between the risks are not weak (Jouini et al., 2004; Molchanov and Cascos, 2016; Feinstein and Rudloff, 2017; Landsman et al., 2016). Furthermore, a vast number of papers introduced and investigated models of risks for the elliptical family of distributions (see, for instance, Owen and Rabinovitch (1983), Valdez and Chernih (2003), Xiao and Valdez (2015), Landsman et al. (2016)). A well-known univariate risk measure is the conditional tail expectation (CTE), CTEq (Y ) := E (Y |F (Y ) ≥ q) , for some risk Y with distribution function F , and some probability level q ∈ (0, 1). In Cousin and Di Bernardino (2014) the authors introduced the following two multivariate versions of the CTE measure: Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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• The lower-orthant CTE at probability level q
1
MCTE q (X) = E (X|F (X) ≥ q) , q ∈ (0, 1).
(9)
• The upper-orthant CTE at probability level q ( ) MCTE q (X) = E X|F (X) ≤ 1 − q , q ∈ (0, 1).
3
(10)
These measures calculate the expected value of a random vector of risks X subject to a condition about its distribution function. The lower-orthant CTE (9) measures the conditional expected value of X subject to the condition that its distribution function, F , is greater than or equal to some probability level q, which directly implies on the case of extreme events of X. The upper-orthant CTE is interpreted in the same way as the lower-orthant measure, only in this case the condition is about the survival function of X, F , which assumes that the survival function of X is below or equal to some probability threshold level 1 − q. Both MCTE q (X) and MCTE q (X) compute that expectation of X around its tail which implies an extreme loss of the risks. Typical values for q are 0.9, 0.95, and 0.99 (see, for example, Shushi (2017)). Another extension of the CTE measure into the multivariate risk measures framework can be found in Landsman et al. (2016), which also derived such extended measure for the elliptical family of distributions. Wang’s premium principle takes the form (Wang, 2002) E Xi e λS
(
πλ (Xi , S) =
E e λS
(
7 8 9 10 11 12 13
15
16 17 18 19
)
λT X
|F (X) ≥ q ) , λi ≥ 0, T E eλ X |F (X) ≥ q
E Xi e
5 6
14
This measure computes the expected value of the risks Xi , i = 1, 2, . . . , n, with a tuned exponential tilting eλS , where S = X1 + X2 + · · · + Xn is the sum of the risks with finite moment generating function (mgf). This measure was computed for the elliptical distributions in Valdez and Chernih (2003), other premium principles for the elliptical distributions can be found in Landsman (2004). The truncated version of Wang’s premium can be defined, as follows:
πq,λ (Xi , S) =
4
)
) , λ ≥ 0.
(
2
20
(
with the tuned exponential tilting eλ X , λ = (λ1 , λ2 , . . . , λn )T ∈ Rn , under the assumption that the distribution function of the risks is greater than or equal to some threshold probability level q ∈ (0, 1), where the mgf exists, i.e., T
(
λT X
MX (λ) := E e
)
< ∞.
(11)
We now derive the truncated Wang’s measure for elliptical random vectors.
πq,λ (X) = µ + Cov (X) λαq,λ − Cov(X)1/2 E δ e
λT X∗
)
βq,λ .
23
25
26
Here αq,λ and βq,λ take the following explicit forms, respectively, MX∗ |(F (X∗ )≥q) (λ) Pr (F (X ) ≥ q)
22
24
Theorem 6. Let X ∽ En (µ, Σ , gn ). Under condition (11), the truncated Wang’s premium measure of X takes the following form
(
21
27
∗
αq,λ =
MX|(F (X)≥q) (λ) Pr (F (X) ≥ q)
,
28
and
29
βq,λ = MX|(F (X)≥q) (λ) , ( T ) where MX|(F (X)≥q) (λ) := E eλ X |F (X) ≥ q is the mgf of X| (F (X) ≥ q) . −1
Proof. Substituting h (X) = eλ
(
E e
λT X
TX
30
31
in Eq. (5), we have
( ) F ∗ (R) ( ) T ∗ (X − µ) |F (X) ≥ q = Cov (X) E ∇ eλ X |F X∗ ≥ q F (R) ( T ∗) 1/2 δ λ X − Cov(X) E e . )
Noticing that F (R) = Pr (F (X) ≥ q), and after some algebraic calculations, we conclude that
(
E eλ
TX
)
X|F (X) ≥ q = Cov (X) λ
MX∗ |(F (X∗ )≥q) (λ) F ∗ (R) F (R)
(
− Cov(X)1/2 E δ eλ
) ∗
TX
(12)
+ µMX∗ |(F (X∗ )≥q) (λ) .
Finally, dividing (12) by MX∗ |(F (X∗ )≥q) (λ), the theorem is then proved. ■ Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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We now derive explicit formulas for the upper and lower orthant MCTE risk measures for the family of elliptical distributions.
3
Theorem 7. The lower and upper orthant MCTE risk measures of X ∽ En (µ, Σ , gn ) take the following forms
1
σZ
MCTE q (X) = µ + √ Cov(X)1/2 E ∇ 1(F (X∗ )≥q) . n Pr (F (X) ≥ q)
4
5
6
7
and
(
σZ
)
(
)
MCTE q (X) = µ + √ ( ) Cov(X)1/2 E ∇ 1(F (X∗ )≤1−q) , n Pr F (X) ≤ 1 − q respectively. Proof. For the lower orthant (9), taking into account Theorem 6, and considering λ = 0, we observe that MX∗ |(F (X)≥q) (0) = 1, F (R) = Pr (F (X) ≥ q) , and βq,0 = 1, so
( ) σZ Cov(X)1/2 E ∇ 1(F (X∗ )≥q) βq,0 nF (R) ( ) Cov(X)1/2 E ∇ 1(F (X∗ )≥q) .
MCTE q (X) = µ + Cov (X) λαq,λ + √
σZ n Pr (F (X) ≥ q)
=µ+ √
For the upper orthant (10), we have E ((X − µ) |F (X) ≤ 1 − q)
= √
σZ
( T ∗ ) ) Cov(X)1/2 E eλ X ∇ 1(F (X∗ )≤1−q) ,
n Pr F (X) ≤ 1 − q
(
and after some algebraic calculations, we finally conclude that MCTE q (X) = E (X|F (X) ≤ 1 − q)
=µ+ √
σZ
( ) ) Cov(X)1/2 E ∇ 1(F (X∗ )≤1−q) .
n Pr F (X) ≤ 1 − q
8
9
10
(
■
Uncited references Landsman and Valdez (2016) Acknowledgments
12
I would like to thank the anonymous referee for the useful comments. This research was supported by the Israel Science Foundation (Grant No. 1686/17).
13
References
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Please cite this article in press as: Shushi T., Stein’s lemma for truncated elliptical random vectors. Statistics and Probability Letters (2018), https://doi.org/10.1016/j.spl.2018.02.008.
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