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## 统计代写|生物统计代写biostatistics代考|Two Special cases

An important special case of SUE distribution is the multivariate unified skewnormal distribution (SUN). Specifically, if the density generator function in (5) is $h^{(m+n)}(u)=(2 \pi)^{-(m+n) / 2} \exp (-u / 2)(u \geq 0)$, we obtain the SUN distribution, denoted by $\mathbf{X} \sim S U N_{n, m}(\theta)$, with pdf (see Arellano-Valle and Azzalini 2006)
$$g_{S U N_{n, m}}(\mathbf{x} ; \boldsymbol{\theta})=\frac{\phi_{n}(\mathbf{x} ; \boldsymbol{\xi}, \boldsymbol{\Omega})}{\Phi_{m}(\boldsymbol{\eta} ; \boldsymbol{\Gamma})} \times \Phi_{m}\left(\eta+\boldsymbol{\Lambda}^{\top} \boldsymbol{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi}) ; \boldsymbol{\Gamma}-\boldsymbol{\Lambda}^{\top} \boldsymbol{\Omega}^{-1} \boldsymbol{\Lambda}\right)$$
where $\phi_{n}(; ; \boldsymbol{\xi}, \boldsymbol{\Omega})$ denotes the pdf of $N_{n}(\xi, \boldsymbol{\Omega})$ and $\Phi_{m}(\cdot ; \boldsymbol{\Gamma})$ denotes the cdf of $N_{m}(\mathbf{0}, \Gamma)$

Another important special case of SUE distribution is the multivariate unified skew-t (SUT) distribution. Specifically, if $h^{(m+n)}(u)=\frac{\Gamma\left(\frac{v+m+n}{2}\right)}{\Gamma\left(\frac{v}{2}\right)(v \pi)^{\frac{m+n}{2}}}\left(1+\frac{u}{v}\right)^{-(v+m+n) / 2}$, we obtain the multivariate unified skew-t (SUT) distribution, denoted by $\mathbf{Y} \sim S U T_{n, m}(\theta, v)$, with pdf
\begin{aligned} &g_{S U T_{n, m}}(\mathbf{x} ; \boldsymbol{\theta}, v)=\frac{t_{n}(\mathbf{x} ; \boldsymbol{\xi}, \boldsymbol{\Omega}, v)}{T_{m}(\eta ; \boldsymbol{\Gamma}, v)} \ &\quad \times T_{m}\left(\eta+\mathbf{\Lambda}^{\top} \mathbf{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi}) ; \frac{v+(\mathbf{x}-\boldsymbol{\xi})^{\top} \mathbf{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi})}{v+n}\left(\boldsymbol{\Gamma}-\boldsymbol{\Lambda}^{\top} \mathbf{\Omega}^{-1} \boldsymbol{\Lambda}\right), v+n\right) \end{aligned}
where $t_{n}(\because ; \xi, \boldsymbol{\Omega}, \nu)$ denotes the pdf of multivariate student- $t$ distribution with location vector $\xi$, dispersion matrix $\boldsymbol{\Omega}$ and degrees of freedom $v$, and $T_{m}(-; \boldsymbol{\Gamma}, \nu)$ denotes the corresponding cdf of $t_{m}(\cdot ; \mathbf{0}, \boldsymbol{\Gamma}, v)$.

Using the results in Lemma 2, we can obtain the marginal and conditional distributions of SUT distribution as presented in the following corollary.

## 统计代写|生物统计代写biostatistics代考|Concomitant Vector of Order Statistics

Let $\mathbf{X}{1}, \ldots, \mathbf{X}{n}$ be random vectors of the same dimension $p$ having elliptical distribution in (1), and let for $i=1,2, \ldots, n, Y_{i}=\mathbf{a}^{\top} \mathbf{X}{i}$ be a linear combination of these vectors, where $\mathbf{a}=\left(a{1}, \ldots, a_{p}\right)^{\top} \in R^{p}$ and $\mathbf{a} \neq \mathbf{0}=(0, \ldots, 0)^{\top}$. Further, let $\mathbf{Y}{(n)}=\left(Y{(1)}, \ldots, Y_{(n)}\right)^{\top}$, with $Y_{(1)}<\cdots<Y_{(n)}$, denote the vector of order statistics arising from $\mathbf{Y}=\left(Y_{1}, \ldots, Y_{n}\right)^{\top}$, and $\mathbf{X}{[r]}$ denote the vector of concomitants corresponding to the $r$ th order statistic $Y{(r)}$.

In this section, we establish a mixture representation for the distribution of $\mathbf{X}{[r]}$ in terms of univariate SUE distribution. For this purpose, we shall first introduce the following notation. Let $1 \leq r \leq n$ be an integer, and for integers $1 \leq j{1}<\cdots<$ $j_{r-1} \leq n-1$, let $\mathbf{S}{j{1} \ldots j_{r-1}}=\operatorname{diag}\left(s_{1}, \ldots, s_{n-1}\right)$ be a $(n-1) \times(n-1)$ diagonal matrix where
$$s_{i}=\left{\begin{array}{cl} 1 & i=j_{1}, \ldots, j_{r-1} \ -1 & \text { otherwise. } \end{array}\right.$$
In the special cases, we have $\mathbf{S}{j{1} \ldots j_{n-1}}=\mathbf{I}{n-1}$ and $\mathbf{S}{j_{0}}=-\mathbf{I}{n-1}$. Furthermore, let, for $i=1, \ldots, n$, the random vector $\mathbf{Y}=\left(Y{1}, \ldots, Y_{n}\right)^{\top}$ be partitioned as
$$\mathbf{Y}=\left(\begin{array}{c} Y_{i} \ \mathbf{Y}{-i} \end{array}\right)$$ where we use the notation $\mathbf{Y}{-i}$ for the vector obtained from $\mathbf{Y}$ by deleting its $i$ th component.
Now, let us introduce the following partitions for $\mu_{\mathbf{Y}}, \Sigma_{\mathbf{Y Y}}$ and $\Sigma_{\mathbf{Y X}}$ :
$$\mu_{\mathbf{Y}}=\left(\begin{array}{c} \mu_{Y_{i}} \ \mu_{\mathbf{Y}{-i}} \end{array}\right), \quad \Sigma{\mathbf{Y Y}}=\left(\begin{array}{cc} \sigma_{Y_{i} Y_{i}} & \sigma_{\mathbf{Y}{-i} Y{i}}^{\top} \ \sigma_{\mathbf{Y}{-i} Y{i}} & \boldsymbol{\Sigma}{\mathbf{Y}{-i} \mathbf{Y}{-i}} \end{array}\right), \quad \boldsymbol{\Sigma}{\mathbf{Y} \mathbf{X}{i}}=\left(\begin{array}{c} \sigma{Y_{i} \mathbf{X}{i}} \ \sigma{\mathbf{Y}{-i} \mathbf{X}{i}} \end{array}\right)$$
We now derive the exact distribution of $\mathbf{X}_{[r]}$, for $r=1,2, \ldots, n$, in the following theorem.

## 统计代写|生物统计代写biostatistics代考|Multivariate Normal Case

In the special case of the multivariate normal distribution, from the general mixture form in Theorem 2 , we get the following corollary.

Corollary 5 If $\left(\begin{array}{c}\boldsymbol{X}{1} \ \vdots \ \boldsymbol{X}{n}\end{array}\right) \sim N_{n p}\left(\boldsymbol{\mu}=\left(\begin{array}{c}\mu_{1} \ \vdots \ \boldsymbol{\mu}{n}\end{array}\right), \boldsymbol{\Sigma}=\left(\begin{array}{ccc}\boldsymbol{\Sigma}{11} & \cdots & \boldsymbol{\Sigma}{1 n} \ \vdots & \ddots & \vdots \ \boldsymbol{\Sigma}{n 1} & \cdots & \boldsymbol{\Sigma}{n n}\end{array}\right)\right)$, then the cdf of $X{[r]}$, for $\boldsymbol{t} \in \mathbb{R}^{p}$, is given by
$$F_{\mathrm{Irl}}(\boldsymbol{t} ; \boldsymbol{\mu}, \boldsymbol{\Sigma})=\sum_{\substack{i=1 \ j_{1}<\ldots<j_{r-1} \leq j_{k} \leq n-1}}^{n} \pi_{i j_{1}, \ldots, j_{r-1}} G_{S U N_{p, n-1}}\left(\boldsymbol{t} ; \theta_{i j_{1}, \ldots, j_{r-1}}\right)$$
where $G_{S U N_{p, n} 1}(\cdot ; \theta)$ denotes the cdf of $S U N_{p, n-1}(\theta)$, and the mixing probabilities are
$$\pi_{i j_{1}, \ldots, j_{r-1}}=\Phi_{n-1}\left(\eta_{i j_{1}, \ldots, j_{r-1}} ; \boldsymbol{\Gamma}{i j{1}, \ldots, j_{r-1}}\right),$$
and $\theta_{i j_{1}, \ldots, j_{r-1}}$ is as given in Theorem $2 .$
In addition, upon using the mean of $S U N_{n, m}(\theta)$ presented in Theorem 1 , the mean of $X_{[r]}$ can be readily obtained as
$$E\left(\boldsymbol{X}{[r]}\right)=\sum{i=1}^{n} \sum_{\substack{j_{1}<\cdots<j_{1}-1 \ 1 \leq j_{k} \leq n-1}} \pi_{i j_{1}, \ldots, j_{r-1}} \mu_{i j_{1}, \ldots, j_{r-1}},$$
where $\mu_{i j_{1}, \ldots, j_{r-1}}$ denotes the mean of $S U N_{p, n-1}\left(\theta_{i j_{1}, \ldots, j_{r-1}}\right)$.

## 统计代写|生物统计代写biostatistics代考|Two Special cases

SUE 分布的一个重要特例是多元统一偏正态分布 (SUN)。具体来说，如果（5) 中的密度生成函数是
$h^{(m+n)}(u)=(2 \pi)^{-(m+n) / 2} \exp (-u / 2)(u \geq 0)$ ，我们得到 SUN 分布，记为 $\mathbf{X} \sim S U N_{n, m}(\theta)$, pdf (见 Arellano-Valle 和 Azzalini 2006)
$$g_{S U N_{n, m}}(\mathbf{x} ; \boldsymbol{\theta})=\frac{\phi_{n}(\mathbf{x} ; \boldsymbol{\xi}, \boldsymbol{\Omega})}{\Phi_{m}(\boldsymbol{\eta} ; \boldsymbol{\Gamma})} \times \Phi_{m}\left(\eta+\boldsymbol{\Lambda}^{\top} \boldsymbol{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi}) ; \boldsymbol{\Gamma}-\mathbf{\Lambda}^{\top} \boldsymbol{\Omega}^{-1} \boldsymbol{\Lambda}\right)$$

SUE 分布的另一个重要特例是多元统一 skew-t (SUT) 分布。具体来说，如果 $h^{(m+n)}(u)=\frac{\Gamma\left(\frac{v+m \mid n}{2}\right)}{\Gamma\left(\frac{v}{2}\right)(v \pi)^{\frac{m-n}{2}}}\left(1+\frac{u}{v}\right)^{-(v+m+n) / 2}$ ，我们获得了 多元统一 skew-t (SUT) 分布，表示为 $\mathbf{Y} \sim S U T_{n, m}(\theta, v)$, 与 $\mathrm{pdf}$
$$g_{S U T n, m}(\mathbf{x} ; \boldsymbol{\theta}, v)=\frac{t_{n}(\mathbf{x} ; \boldsymbol{\xi}, \boldsymbol{\Omega}, v)}{T_{m}(\eta ; \boldsymbol{\Gamma}, v)} \quad \times T_{m}\left(\eta+\mathbf{\Lambda}^{\top} \mathbf{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi}) ; \frac{v+(\mathbf{x}-\boldsymbol{\xi})^{\top} \boldsymbol{\Omega}^{-1}(\mathbf{x}-\boldsymbol{\xi})}{v+n}\left(\mathbf{\Gamma}-\mathbf{\Lambda}^{\top} \boldsymbol{\Omega}^{-1} \boldsymbol{\Lambda}\right), v+n\right)$$

## 统计代写|生物统计代写biostatistics代考|Multivariate Normal Case

$$F_{\mathrm{Irl}}(\boldsymbol{t} ; \boldsymbol{\mu}, \boldsymbol{\Sigma})=\sum_{i=1}^{n} \sum_{j_{1}<\ldots<j_{r-1} \leq j_{k} \leq n-1} \pi_{i j_{1}, \ldots, j_{r-1}} G_{S U N_{p n-1}}\left(\boldsymbol{t} ; \theta_{i j_{1}, \ldots, j_{r-1}}\right)$$

$$\pi_{i j_{1}, \ldots, j_{r-1}}=\Phi_{n-1}\left(\eta_{i j_{1}, \ldots, j_{r-1}} ; \boldsymbol{\Gamma} i j 1, \ldots, j_{r-1}\right),$$

$$E(\boldsymbol{X}[r])=\sum i=1^{n} \sum_{j_{1}<\cdots<j_{1}-1 \leq j j_{k} \leq n-1} \pi_{i j_{1}, \ldots, j_{r-1}} \mu_{i j_{1}, \ldots, j_{r-1}},$$

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