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Multivariate Doubly Truncated Moments for a Class of Multivariate Location-Scale Mixture of Elliptical Distributions

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Abstract

In this paper, we investigate multivariate doubly truncated moments for a class of multivariate location-scale mixture of elliptical (LSME) distributions. This rich family includes some well-known distributions, such as location-scale mixture of normal, location-scale mixture of Student-\(t\), location-scale mixture of logistic and location-scale mixture of Laplace distributions, as special cases. We first present general formulae for computing the first two moments of the LSME distributions under the double truncation. We then consider a special case for cross moment. As an application, we present the results of multivariate tail conditional expectation (MTCE) for generalized hyperbolic (MGH) distribution.

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ACKNOWLEDGMENTS

The research was supported by the National Natural Science Foundation of China (nos. 12071251, 11571198).

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Correspondence to Chuancun Yin.

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The authors declare that there is no conflict of interests regarding the publication of this article.

APPENDIX

APPENDIX

In this appendix, we provide all the proofs of the main results, states in Section 3, of this paper.

Proof of Theorem 1. (1) Using the transformation \(\mathbf{Z}=(\theta\boldsymbol{\Sigma})^{-\frac{1}{2}}(\mathbf{Y}-\boldsymbol{\mu}-\theta\boldsymbol{\beta})\), we have

$${\textrm{E}}[{\mathbf{Y}}|{a}<{\mathbf{Y}}\leq{b}]=\frac{1}{\alpha_{1,n}}\int\limits_{a}^{b}{y}f_{\mathbf{Y}}(y)\textrm{d}y=\frac{1}{\alpha_{1,n}}\int\limits_{a}^{b}\int\limits_{\Omega_{\theta}}yf_{\mathbf{Y}|\Theta}(y)f_{\Theta}(\theta){\textrm{d}}\theta{\textrm{d}}{y}$$
$${}=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\int\limits_{a}^{b}{y}\frac{c_{1,n}}{\sqrt{|\theta\boldsymbol{\Sigma}|}},$$
$$g_{n}\left(\frac{1}{2}(y-\mu-\theta\boldsymbol{\beta})^{T}(\theta\boldsymbol{\Sigma})^{-1}(y-\mu-\theta\boldsymbol{\beta})\right)\textrm{d}y\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\!\int\limits_{\Omega_{\theta}}\!f_{\Theta}(\theta)\int\limits_{a^{\prime}}^{b^{\prime}}\!\left(\left(\theta\boldsymbol{\Sigma}\right)^{1/2}\mathbf{z}\!+\boldsymbol{\mu}\!+\theta\boldsymbol{\beta}\right)c_{1,n}g_{n}\left(\frac{1}{2}{\mathbf{z}}^{T}\mathbf{z}\right)\textrm{d}\mathbf{z}\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\int\limits_{a^{\prime}}^{b^{\prime}}\left(\left(\theta\boldsymbol{\Sigma}\right)^{1/2}\mathbf{z}c_{1,n}g_{n}\{\frac{1}{2}\mathbf{z}^{T}\mathbf{z}\}+(\boldsymbol{\mu}+\theta\boldsymbol{\beta})c_{1,n}g_{n}\left(\frac{1}{2}\mathbf{z}^{T}\mathbf{z}\right)\right)\textrm{d}\mathbf{z}\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\boldsymbol{\Sigma}^{1/2}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\theta^{1/2}\boldsymbol{\delta}\textrm{d}\theta+\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)(\boldsymbol{\mu}+\theta\boldsymbol{\beta})\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\boldsymbol{\Sigma}^{1/2}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\theta^{1/2}\boldsymbol{\delta}\textrm{d}\theta+\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\theta\boldsymbol{\beta}\textrm{d}\theta+\boldsymbol{\mu},$$

where \(a^{\prime}=(a^{\prime}_{1},a^{\prime}_{2},...,a^{\prime}_{n})^{T}=(\theta\boldsymbol{\Sigma})^{-\frac{1}{2}}(a-\boldsymbol{\mu}-\theta\boldsymbol{\beta}),b^{\prime}=(b^{\prime}_{1},b^{\prime}_{2},...,b^{\prime}_{n})^{T}=(\theta\boldsymbol{\Sigma})^{-\frac{1}{2}}(b-\boldsymbol{\mu}-\theta\boldsymbol{\beta})\), and \(\boldsymbol{\delta}=(\delta_{1},\delta_{2},...,\delta_{n})^{T}.\)

Note that

$$\delta_{k}=c_{1,n}\int\limits_{a^{\prime}_{-k}}^{b^{\prime}_{-k}}\int\limits_{a^{\prime}_{k}}^{b^{\prime}_{k}}z_{k}g_{n}\left(\frac{1}{2}\mathbf{z}_{-k}^{T}\mathbf{z}_{-k}+\frac{1}{2}z_{k}^{2}\right)\textrm{d}z_{k}\textrm{d}\mathbf{z}_{-k}=c_{1,n}\int\limits_{a^{\prime}_{-k}}^{b^{\prime}_{-k}}\int\limits_{a^{\prime}_{k}}^{b^{\prime}_{k}}-\partial\overline{G}_{2,n}\left(\frac{1}{2}\mathbf{z}_{-k}^{T}\mathbf{z}_{-k}+\frac{1}{2}z_{k}^{2}\right)\textrm{d}z_{k}\textrm{d}\mathbf{z}_{-k}$$
$${}={c_{1,n}}\int\limits_{a^{\prime}_{-k}}^{b^{\prime}_{-k}}\left(\overline{G}_{2,n}^{a^{\prime}_{k}}\left(\frac{1}{2}\mathbf{z}_{-k}^{T}\mathbf{z}_{-k}+\frac{1}{2}a_{k}^{\prime 2}\right)-\overline{G}_{2,n}^{b^{\prime}_{k}}\left(\frac{1}{2}\mathbf{z}_{-k}^{T}\mathbf{z}_{-k}+\frac{1}{2}b_{k}^{\prime 2}\right)\right)\textrm{d}\mathbf{z}_{-k}$$
$${}=c_{1,n}\left(\frac{\alpha_{2,n-1}^{a^{\prime}_{k}}}{c_{2,n-1}^{a^{\prime}_{k}}}-\frac{\alpha_{2,n-1}^{b^{\prime}_{k}}}{c_{2,n-1}^{b^{\prime}_{k}}}\right).$$

(2) Similarly, using the transformation \(\mathbf{Z}=(\theta\Sigma)^{-\frac{1}{2}}(\mathbf{Y}-\mu-\theta\beta)\), we have

$${\textrm{E}}\left[\mathbf{Y}\mathbf{Y}^{T}|{a}<\mathbf{Y}\leq{b}\right]=\frac{1}{\alpha_{1,n}}\int\limits_{{a}}^{{b}}y{y}^{T}f_{\mathbf{Y}}(y)\textrm{d}{y}=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\int\limits_{{a}}^{{b}}{y}{y}^{T}\frac{c_{1,n}}{\sqrt{|\theta\Sigma|}},$$
$$g_{n}\left(\frac{1}{2}(y-\mu-\theta\beta)^{T}(\theta\Sigma)^{-1}(y-\mu-\theta\beta)\right)\textrm{d}y\textrm{d}\theta$$
$${}=\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\left((\theta\Sigma)^{1/2}{\textrm{E}}[\mathbf{ZZ}^{T}|a^{\prime}<\mathbf{Z}\leq b^{\prime}](\theta\Sigma)^{1/2}\right.+(\theta\Sigma)^{1/2}\!{\textrm{E}}[\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}](\mu+\theta\beta)^{T}$$
$${}+(\mu+\theta\beta){\textrm{E}}[\mathbf{Z}^{T}|a^{\prime}<\mathbf{Z}\leq b^{\prime}](\theta\Sigma)^{1/2}+\left.\theta\mu\beta^{T}+\theta\beta\mu^{T}+\theta^{2}\beta\beta^{T}\right)\textrm{d}\theta+\mu\mu^{T}.$$

For \({\textrm{E}}[\mathbf{Z}\mathbf{Z}^{T}|a^{\prime}<\mathbf{Z}\leq a^{\prime}],\) using the definition of conditional expectation we obtain

$${\textrm{E}}[\mathbf{Z}\mathbf{Z}^{T}|a^{\prime}<\mathbf{Z}\leq a^{\prime}]=\frac{1}{\alpha_{1,n}}\int\limits_{a^{\prime}}^{b^{\prime}}zz^{T}c_{1,n}g_{n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z$$
$${}=\frac{1}{\alpha_{1,n}}\boldsymbol{\Omega},$$

where \(\Omega=(\Omega_{i,j})_{i,j=1}^{n}.\)

Note that, for \(i=j,\) we have

$$\Omega_{ii}=\int\limits_{a^{\prime}}^{b^{\prime}}z_{i}^{2}c_{1,n}g_{n}\left(\frac{1}{2}{z}{z}^{T}\right)\textrm{d}\mathbf{z}=\int\limits_{a^{\prime}_{-i}}^{b^{\prime}_{-i}}\int\limits_{a^{\prime}_{i}}^{b^{\prime}_{i}}-z_{i}c_{1,n}\partial_{i}\overline{G}_{2,n}\left\{\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}z_{i}^{2}\right\}\textrm{d}z_{i}\textrm{d}\textbf{z}_{-i}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{-i}}^{b^{\prime}_{-i}}\left(a^{\prime}_{i}\overline{G}_{2,n}^{a^{\prime}_{i}}\left(\frac{1}{2}z_{-i}^{T}z_{-i}+\frac{1}{2}a_{i}^{\prime 2}\right)-b^{\prime}_{i}\overline{G}_{2,n}^{b^{\prime}_{i}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{i}^{\prime 2}\right)\right.$$
$${}+\left.\int\limits_{a^{\prime}}^{b^{\prime}}\overline{G}_{2,n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z_{i}\right)\textrm{d}z_{-i}$$
$${}=\int\limits_{a^{\prime}_{-i}}^{b^{\prime}_{-i}}\left(a^{\prime}_{i}\frac{c_{1,n}}{c_{2,n-1}^{a^{\prime}_{i}}}c_{2,n-1}^{a^{\prime}_{i}}\overline{G}_{2,n}^{a^{\prime}_{i}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}-\left.b^{\prime}_{i}\frac{c_{1,n}}{c_{2,n-1}^{b^{\prime}_{i}}}c_{2,n-1}^{b^{\prime}_{i}}\overline{G}_{2,n}^{b^{\prime}_{i}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{-i}+\int\limits_{a^{\prime}}^{b^{\prime}}c_{1n}\overline{G}_{2,n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z$$
$${}=\frac{c_{1,n}}{c_{2,n-1}^{a^{\prime}_{i}}}a^{\prime}_{i}\alpha_{2,n-1}^{a^{\prime}_{i}}-\frac{c_{1,n}}{c_{2,n-1}^{b^{\prime}_{i}}}b^{\prime}_{i}\alpha_{2,n-1}^{b^{\prime}_{i}}+\frac{c_{1,n}}{c_{2,n}}\alpha_{2,n},$$

while \(i\neq j,\)

$$\Omega_{ij}=\int\limits_{a^{\prime}}^{b^{\prime}}z_{i}z_{j}c_{1,n}g_{n}\left(\frac{1}{2}{z}{z}^{T}\right)\textrm{d}{z}=\int\limits_{a^{\prime}_{-i}}^{b^{\prime}_{-i}}\int\limits_{a^{\prime}_{i}}^{b^{\prime}_{i}}-z_{j}c_{1,n}\partial_{i}\overline{G}_{2,n}\left(\frac{1}{2}\mathbf{z}_{-i}\mathbf{z}_{-i}^{T}+\frac{1}{2}z_{i}^{2}\right)\textrm{d}z_{i}\textrm{d}{z}_{-i}$$
$${}=\int\limits_{a^{\prime}_{-i}}^{b^{\prime}_{-i}}\!\left(z_{j}c_{1,n}\overline{G}_{2,n-1}^{a^{\prime}_{i}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\!\frac{1}{2}a_{i}^{\prime 2}\right)\right.-\left.z_{j}c_{1,n}\overline{G}_{2,n-1}^{b^{\prime}_{i}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{-i}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{-i,-j}}^{b^{\prime}_{-i,-j}}\int\limits_{a^{\prime}_{j}}^{b^{\prime}_{j}}\left(z_{j}\overline{G}_{2,n-1}^{a^{\prime}_{i}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}-\left.z_{j}\overline{G}_{2,n-1}^{b^{\prime}_{i}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{j}\textrm{d}z_{-i,-j}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{-i,-j}}^{b^{\prime}_{-i,-j}}\int\limits_{a^{\prime}_{j}}^{b^{\prime}_{j}}\left(-\partial_{j}\overline{G}_{3,n-1}^{a^{\prime}_{i}}\left(\frac{1}{2}\textbf{z}_{-i,-j}\textbf{z}_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}+\left.\partial_{j}\overline{G}_{3,n-1}^{b^{\prime}_{i}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{j}\textrm{d}z_{-i,-j}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{-i,-j}}^{b^{\prime}_{-i,-j}}\left(\overline{G}_{3,n-2}^{a^{\prime}_{i},a^{\prime}_{j}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}a_{j}^{\prime 2}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}-\left.\overline{G}_{3,n-2}^{a^{\prime}_{i},b^{\prime}_{i}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}a_{i}^{\prime 2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right.+\left.\overline{G}_{3,n-2}^{b^{\prime}_{i},b^{\prime}_{j}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}b_{j}^{\prime 2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right.$$
$${}-\left.\overline{G}_{3,n-2}^{a^{\prime}_{j},b^{\prime}_{i}}\left(\frac{1}{2}z_{-i,-j}z_{-i,-j}^{T}+\frac{1}{2}a_{j}^{\prime 2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{-i,-j}$$
$${}=\frac{{c_{1,n}}}{c_{3,n-2}^{a^{\prime}_{j}a^{\prime}_{i}}}\alpha_{3,n-2}^{a^{\prime}_{j}a^{\prime}_{i}}-\frac{{c_{1,n}}}{c_{3,n-2}^{b^{\prime}_{j}a^{\prime}_{i}}}\alpha_{3,n-2}^{b^{\prime}_{j}a^{\prime}_{i}}+\frac{{c_{1,n}}}{c_{3,n-2}^{b^{\prime}_{j}b^{\prime}_{i}}}\alpha_{3,n-2}^{b^{\prime}_{j}b^{\prime}_{i}}-\frac{{c_{1,n}}}{c_{3,n-2}^{a^{\prime}_{j}b^{\prime}_{i}}}\alpha_{3,n-2}^{a^{\prime}_{j}b^{\prime}_{i}},$$

where we have used integration by parts in the third and fifth equalities.

As for \({\textrm{E}}[\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}],\) using (1) we directly obtain

$${\textrm{E}}[\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]=\frac{\boldsymbol{\delta}}{\alpha_{1,n}}.$$

Consequently, we obtain (7), ending the proof of (2).

Proof of Remark 1.

$${\textrm{E}}[\mathbf{YY^{T}Y}|a<\mathbf{Y}\leq b]=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega}f(\theta)\int\limits_{a^{\prime}}^{b^{\prime}}\left[\left(\theta\Sigma\right)^{1/2}\mathbf{z}+\mu+\theta\beta\right]$$
$$\left[\mathbf{z}^{T}\left(\theta\Sigma\right)^{1/2}+\mu^{T}+\theta\beta^{T}\right]\left[\left(\theta\Sigma\right)^{1/2}\mathbf{z}+\mu+\theta\beta\right]\textrm{d}\mathbf{z}\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega}f(\theta)\int\limits_{a^{\prime}}^{b^{\prime}}\left[(\theta\Sigma)^{1/2}\mathbf{z}\mathbf{z}^{T}(\theta\Sigma)^{1/2}(\theta\Sigma)^{1/2}\mathbf{z}\right.$$
$${}+(\theta\Sigma)^{1/2}\mathbf{z}\mathbf{z}^{T}(\theta\Sigma)^{1/2}(\mu+\theta\beta)$$
$${}+(\theta\Sigma)^{1/2}\mathbf{z}\mu^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)+(\theta\Sigma)^{1/2}\mathbf{z}(\theta\beta)^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)$$
$${}+\mu\mathbf{z}^{T}(\theta\Sigma)^{1/2}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)+\mu\mu^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)$$
$${}+\theta\mu\beta^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)+\theta\beta\mathbf{z}^{T}(\theta\Sigma)^{1/2}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)$$
$${}+\theta\beta\mu^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)+\left.\theta^{2}\beta\beta^{T}\left((\theta\Sigma)^{1/2}\mathbf{z}+\mu+\theta\beta\right)\right]\textrm{d}\mathbf{z}\textrm{d}\theta$$
$${}=\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\left[\theta^{3/2}\Sigma^{1/2}\textrm{E}[\mathbf{Z}\mathbf{Z}^{T}\Sigma\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]\right.+(\theta\Sigma)^{1/2}\textrm{E}[\mathbf{Z}\mathbf{Z}^{T}|a^{\prime}<\mathbf{Z}\leq b^{\prime}](\theta\Sigma)^{1/2}(\mu+\theta\beta)$$
$${}+(\theta\Sigma)^{1/2}\textrm{E}[\mathbf{Z}\mu^{T}(\theta\Sigma)^{1/2}\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]+(\theta\Sigma)^{1/2}\textrm{E}[\mathbf{Z}(\theta\beta)^{T}(\theta\Sigma)^{1/2}\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]$$
$${}+\theta(\mu+\beta)\textrm{E}[\mathbf{Z}^{T}\Sigma\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]+(\theta\Sigma)^{1/2}\textrm{E}[\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]\left(\mu^{T}+(\theta\beta)^{T}\right)(\mu+\theta\beta)$$
$${}+\left(\mu\mu^{T}+\theta\mu\beta^{T}+\theta\beta\mu^{T}+\theta^{2}\beta\beta^{T}\right)(\theta\Sigma)^{1/2}\textrm{E}[\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]$$
$${}+(\mu+\theta\beta)\textrm{E}[\mathbf{Z}^{T}|a^{\prime}<\mathbf{Z}\leq b^{\prime}](\theta\Sigma)^{1/2}(\mu+\theta\beta)$$
$${}+\left.\left(\mu\mu^{T}+\theta\mu\beta^{T}+\theta\beta\mu^{T}+\theta^{2}\beta\beta^{T}\right)(\mu+\theta\beta)\right]\textrm{d}\theta.$$

As for \(\textrm{E}[\mathbf{Z}\mathbf{Z}^{T}\Sigma\mathbf{Z}|a^{\prime}<\mathbf{Z}\leq b^{\prime}]\), its result is an \(n\)-dimensional vector formed by a linear combination of the following three formulas.For \(i\neq j\neq k,\)

$$\int\limits_{a^{\prime}}^{b^{\prime}}z_{i}z_{j}z_{k}c_{1,n}g_{n}\left(\frac{1}{2}\mathbf{z}\mathbf{z}^{T}\right)\textrm{d}\mathbf{z}$$
$${}=c_{1,n}\int\limits_{a_{-i,-j}^{\prime}}^{b_{-i,-j}^{\prime}}\int\limits_{a_{j}}^{b_{j}^{\prime}}\left(-z_{k}\partial_{j}\overline{G}_{3,n-1}^{a_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i,-j}\mathbf{z}_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}+\left.z_{k}\partial_{j}\overline{G}_{3,n-1}^{b_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i,-j}\mathbf{z}_{-i,-j}^{T}+\frac{1}{2}z_{j}^{2}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}\textrm{z}_{j}\textrm{d}\mathbf{z}_{-i,-j}$$
$${}=\frac{c_{1,n}}{c_{4,n-3}^{a_{i}^{\prime}a_{j}^{\prime}a_{k}^{\prime}}}\alpha_{4,n-3}^{a_{i}^{\prime}a_{j}^{\prime}a_{k}^{\prime}}-\frac{c_{1,n}}{c_{4,n-3}^{a_{i}^{\prime}a_{j}^{\prime}b_{k}^{\prime}}}\alpha_{4,n-3}^{a_{i}^{\prime}a_{j}^{\prime}b_{k}^{\prime}}+\frac{c_{1,n}}{c_{4,n-3}^{a_{i}^{\prime}b_{j}^{\prime}b_{k}^{\prime}}}\alpha_{4,n-3}^{a_{i}^{\prime}b_{j}^{\prime}b_{k}^{\prime}}-\frac{c_{1,n}}{c_{4,n-3}^{a_{i}^{\prime}b_{j}^{\prime}a_{k}^{\prime}}}\alpha_{4,n-3}^{a_{i}^{\prime}b_{j}^{\prime}a_{k}^{\prime}},$$

while \(i=k\neq j\),

$$\int\limits_{a^{\prime}}^{b^{\prime}}z_{i}^{2}z_{j}c_{1,n}g_{n}\left(\frac{1}{2}\mathbf{z}z^{T}\right)\textrm{d}\mathbf{z}$$
$${}=c_{1,n}\int\limits_{a_{-i,-j}^{\prime}}^{b_{-i,-j}^{\prime}}\int\limits_{a_{j}^{\prime}}^{b_{j}^{\prime}}\left(a_{i}^{\prime}z_{j}\overline{G}_{2,n-1}^{a_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i}\mathbf{z}_{-i}^{T}+\frac{1}{2}a_{i}^{\prime 2}\right)\right.$$
$${}-\left.b_{i}^{\prime}z_{j}\overline{G}_{2,n-1}^{b_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i}\mathbf{z}_{-i}^{T}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}z_{-j}\textrm{d}\mathbf{z}_{-i,-j}+c_{1,n}\int\limits_{a^{\prime}}^{b^{\prime}}z_{j}\overline{G}_{2,n}\left(\frac{1}{2}\mathbf{zz^{T}}\right)\textrm{d}\mathbf{z}$$
$${}=a_{i}^{\prime}\frac{c_{1,n}}{c_{3,n-2}^{a_{i}^{\prime}a_{j}^{\prime}}}\alpha_{3,n-2}^{a_{i}^{\prime}a_{j}^{\prime}}-b_{i}^{\prime}\frac{c_{1,n}}{c_{3,n-2}^{a_{i}^{\prime}a_{j}^{\prime}}}\alpha_{3,n-2}^{a_{i}^{\prime}a_{j}^{\prime}}+b_{i}^{\prime}\frac{c_{1,n}}{c_{3,n-2}^{b_{i}^{\prime}b_{j}^{\prime}}}\alpha_{3,n-2}^{b_{i}^{\prime}b_{j}^{\prime}}$$
$${}-b_{i}^{\prime}\frac{c_{1,n}}{c_{3,n-2}^{b_{i}^{\prime}a_{j}^{\prime}}}\alpha_{3,n-2}^{b_{i}^{\prime}a_{j}^{\prime}}+\frac{c_{1,n}}{c_{3,n-1}^{a_{j}^{\prime}}}\alpha_{3,n-1}^{a_{j}^{\prime}}-\frac{c_{1,n}}{c_{3,n-1}^{b_{j}^{\prime}}}\alpha_{3,n-1}^{b_{j}^{\prime}},$$

while \(i=j=k\),

$$\int\limits_{a^{\prime}}^{b^{\prime}}z_{i}^{3}c_{1,n}g_{n}\left(\frac{1}{2}\mathbf{zz^{T}}\right)\textrm{d}\mathbf{z}=c_{1,n}\int\limits_{a_{-i}^{\prime}}^{b_{-i}^{\prime}}\left(a_{i}^{\prime 2}\overline{G}_{2,n-1}^{a_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i}\mathbf{z}_{-i}^{T}+\frac{1}{2}a_{i}^{\prime 2}\right){}\right.$$
$${}\left.-b_{i}^{\prime 2}\overline{G}_{2,n-1}^{b_{i}^{\prime}}\left(\frac{1}{2}\mathbf{z}_{-i}\mathbf{z}_{-i}^{T}+\frac{1}{2}b_{i}^{\prime 2}\right)\right)\textrm{d}\mathbf{z}_{-i}+c_{1,n}\int\limits_{a^{\prime}}^{b^{\prime}}2z_{i}\overline{G}_{2,n}\left(\frac{1}{2}\mathbf{zz^{T}}\right)\textrm{d}\mathbf{z}.$$

Proof of Theorem 2. Using the transformation \(\mathbf{Z}=(\theta\Sigma)^{1/2}(\mathbf{Y}-\mu_{d})\), we have

$${\textrm{E}}_{\alpha_{1,n}}(\widetilde{\mathbf{Y}}_{d}^{\mathbf{k}})=\frac{1}{\alpha_{1,n}}\int\limits_{a_{d}}^{b_{d}}{\widetilde{y}}_{d}^{\mathbf{k}}f_{\mathbf{Y}}(y)\textrm{d}y=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\int\limits_{a_{d}}^{b_{d}}(\widetilde{y}_{d})^{\mathbf{k}}f_{\mathbf{Y}|\Theta}(y|\Theta)\textrm{d}y\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\!\int\limits_{\Omega_{\theta}}\!f_{\Theta}(\theta)\!\int\limits_{a_{d}}^{b_{d}}\!(\widetilde{y}_{d})^{\mathbf{k}}\!\frac{c_{1,n}}{\sqrt{\theta\Sigma}}\!g_{n}\left(\frac{1}{2}({y\!-\mu_{d}})^{T}(\theta\Sigma)^{-1}({y\!-\mu_{d}})\right)\textrm{d}y\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)\int\limits_{a^{\prime}_{d}}^{b^{\prime}_{d}}\left((\theta\Sigma)^{1/2}z+\mu+\theta\beta\right)^{\mathbf{k}}c_{1,n}g_{n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}C_{k}^{l}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)(\theta\Sigma)^{\frac{l}{2}}(\mu+\theta\beta)^{k-l}\int\limits_{a^{\prime}_{d}}^{b^{\prime}_{d}}z^{l}c_{1,n}g_{n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z\textrm{d}\theta$$
$${}=\frac{1}{\alpha_{1,n}}C_{k}^{l}\int\limits_{\Omega_{\theta}}f_{\Theta}(\theta)(\theta\Sigma)^{\frac{l}{2}}(\mu+\theta\beta)^{k-l}\Lambda\textrm{d}\theta,$$

where \(\Lambda=(\Lambda_{1},\Lambda_{2},...,\Lambda_{n})^{T}\).

Note that

$$\Lambda_{i}=\int\limits_{a^{\prime}_{d}}^{b^{\prime}_{d}}z_{i}^{l}c_{1,n}g_{n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z=c_{1,n}\int\limits_{a^{\prime}_{d-i}}^{b^{\prime}_{d-i}}\int\limits_{a^{\prime}_{di}}^{b^{\prime}_{di}}-z_{i}^{l-1}\partial_{i}\overline{G}_{2,n}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}z_{i}^{2}\right)\textrm{d}z_{i}\textrm{d}z_{-i}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{d-i}}^{b^{\prime}_{d-i}}\left[\left({a^{\prime}_{di}}^{l-1}\overline{G}_{2,n}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-{b^{\prime}_{di}}^{l-1}\overline{G}_{2,n}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)\right.$$
$${}+\left.(l-1)\int\limits_{a^{\prime}_{di}}^{b^{\prime}_{di}}z_{i}^{l-2}\overline{G}_{2,n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z_{i}\right]\textrm{d}z_{-i}$$
$${}=c_{1,n}\int\limits_{a^{\prime}_{d-i}}^{b^{\prime}_{d-i}}\left[\left({a^{\prime}_{di}}^{l-1}\overline{G}_{2,n}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-{b^{\prime}_{di}}^{l-1}\overline{G}_{2,n}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)\right.$$
$${}+(l-1)\left({a^{\prime}_{di}}^{l-3}\overline{G}_{3,n}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-{b^{\prime}_{di}}^{l-3}\overline{G}_{3,n}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)$$
$${}+\left.(l-1)(l-3)\int\limits_{a^{\prime}_{di}}^{b^{\prime}_{di}}z_{i}^{l-4}\overline{G}_{3,n}\left(\frac{1}{2}zz^{T}\right)\textrm{d}z_{i}\right]\textrm{d}z_{-i}.$$

By summarizing, the result can be divided into odd and even cases.

When \(l\) is odd,

$$\Lambda_{i}=c_{1,n}\int\limits_{a^{\prime}_{d-i}}^{b^{\prime}_{d-i}}\left[a_{di}^{\prime l-1}\overline{G}_{2,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-b_{di}^{\prime l-1}\overline{G}_{2,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right.$$
$${}+(l-1)\left(a_{di}^{\prime l-3}\overline{G}_{3,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-b_{di}^{\prime l-1}\overline{G}_{3,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)$$
$${}+...$$
$${}+(l-1)(l-3)...\left(l-(2m-3)\right)\left(a_{di}^{\prime(l-(2m-1))}\overline{G}_{m+1,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)\right.$$
$${}-\left.b_{di}^{\prime(l-(2m-1))}\overline{G}_{m+1,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)$$
$${}+...$$
$${}+(l-1)(l-3)...1\left(\overline{G}_{\frac{l+3}{2},n-1}^{a_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)\right.$$
$${}-\left.\left.\overline{G}_{\frac{l+3}{2},n-1}^{b_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)\right]\textrm{d}z_{-i}$$
$${}=\left(a_{di}^{\prime l-1}\frac{c_{1,n}}{c_{2,n-1}^{a^{\prime}_{di}}}\alpha_{2,n-1}^{a^{\prime}_{di}}-b_{di}^{\prime l-1}\frac{c_{1,n}}{c_{2,n-1}^{b^{\prime}_{di}}}\alpha_{2,n-1}^{b^{\prime}_{di}}\right)$$
$${}+(l-1)\left(a_{di}^{\prime l-3}\frac{c_{1,n}}{c_{3,n-1}^{a^{\prime}_{di}}}\alpha_{3,n-2}^{a^{\prime}_{di}}-b_{di}^{\prime l-3}\frac{c_{1,n}}{c_{3,n-1}^{b^{\prime}_{di}}}\alpha_{3,n-1}^{b^{\prime}_{di}}\right)$$
$${}+...+(l-1)(l-3)...(l-(2m-3))$$
$${}\times\left(a_{di}^{\prime l-(2m-1)}\frac{c_{1,n}}{c_{m+1,n-1}^{a^{\prime}_{di}}}\alpha_{m+1,n-1}^{a^{\prime}_{di}}-b_{di}^{\prime l-(2m-1)}\frac{c_{1,n}}{c_{m+1,n-1}^{b^{\prime}_{di}}}\alpha_{m+1,n-1}^{b^{\prime}_{di}}\right)+...$$
$${}+(l-1)(l-3)...1\left(\frac{c_{1,n}}{c_{\frac{l+3}{2},n-1}^{a^{\prime}_{di}}}\alpha_{\frac{l+3}{2},n-1}^{a^{\prime}_{di}}-\frac{c_{1,n}}{c_{\frac{l+3}{2},n-1}^{b^{\prime}_{di}}}\alpha_{\frac{l+3}{2},n-1}^{b^{\prime}_{di}}\right).$$

When \(l\) is even,

$$\Lambda_{i}=c_{1,n}\int\limits_{a^{\prime}_{d-i}}^{b^{\prime}_{d-i}}\left[a_{di}^{\prime l-1}\overline{G}_{2,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)-b_{di}^{\prime l-1}\overline{G}_{2,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right.$$
$${}+(l-1)\left(a_{di}^{\prime l-3}\overline{G}_{3,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)\right.$$
$${}-\left.b_{di}^{\prime l-1}\overline{G}_{2,n-1}^{b^{\prime}_{di}}\overline{G}_{2,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)$$
$${}+...$$
$${}+(l-1)(l-3)...\left(l-(2m-3)\right)\left(a_{di}^{\prime(l-(2m-1))}\overline{G}_{m+1,n-1}^{a^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)\right.$$
$${}-\left.b_{di}^{\prime(l-(2m-1))}\overline{G}_{m+1,n-1}^{b^{\prime}_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)$$
$${}+...$$
$${}+(l-1)(l-3)...3\left(\overline{G}_{\frac{l}{2}+1,n-1}^{a_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}a_{di}^{\prime 2}\right)\right.$$
$${}-\left.\left.\overline{G}_{\frac{l}{2}+1,n-1}^{b_{di}}\left(\frac{1}{2}z_{-i}z_{-i}^{T}+\frac{1}{2}b_{di}^{\prime 2}\right)\right)\right]\textrm{d}z_{-i}+(l-1)(l-2)...1c_{1,n}\frac{\alpha_{\frac{l}{2}+1,n}}{c_{\frac{l}{2}+1,n}}$$
$${}=\left(a_{di}^{\prime l-1}\frac{c_{1,n}}{c_{2,n-1}^{a^{\prime}_{di}}}\alpha_{2,n-1}^{a^{\prime}_{di}}-b_{di}^{\prime l-1}\frac{c_{1,n}}{c_{2,n-1}^{b^{\prime}_{di}}}\alpha_{2,n-1}^{b^{\prime}_{di}}\right)$$
$${}+(l-1)\left(a_{di}^{\prime l-3}\frac{c_{1,n}}{c_{3,n-1}^{a^{\prime}_{di}}}\alpha_{3,n-2}^{a^{\prime}_{di}}-b_{di}^{\prime l-3}\frac{c_{1,n}}{c_{3,n-1}^{b^{\prime}_{di}}}\alpha_{3,n-1}^{b^{\prime}_{di}}\right)$$
$${}+...+(l-1)(l-3)...(l-(2m-3))$$
$${}\times\left(a_{di}^{\prime l-(2m-1)}\frac{c_{1,n}}{c_{m+1,n-1}^{a^{\prime}_{di}}}\alpha_{m+1,n-1}^{a^{\prime}_{di}}-b_{di}^{\prime l-(2m-1)}\frac{c_{1,n}}{c_{m+1,n-1}^{b^{\prime}_{di}}}\alpha_{m+1,n-1}^{b^{\prime}_{di}}\right)+...$$
$${}+(l-1)(l-3)...3\left(\frac{c_{1,n}}{c_{\frac{l}{2}+1,n-1}^{a^{\prime}_{di}}}\alpha_{\frac{l}{2}+1,n-1}^{a^{\prime}_{di}}-\frac{c_{1,n}}{c_{\frac{l}{2}+1,n-1}^{b^{\prime}_{di}}}\alpha_{\frac{l}{2}+1,n-1}^{b^{\prime}_{di}}\right)$$
$${}+(l-1)(l-2)...1c_{1,n}\frac{\alpha_{\frac{l}{2}+1,n}}{c_{\frac{l}{2}+1,n}}{.}$$

This completes the proof of Theorem 2.

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Han, X., Yin, C. Multivariate Doubly Truncated Moments for a Class of Multivariate Location-Scale Mixture of Elliptical Distributions. Math. Meth. Stat. 32, 155–175 (2023). https://doi.org/10.3103/S1066530723030043

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