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## 统计代写|生物统计代写biostatistics代考|Joint Distribution

We now derive the exact joint distribution of $\left(X_{[r]}, Y_{(r)}\right)^{\top}$, for $r=1, \ldots, n$, and show that this distribution is indeed a mixture of unified skew-elliptical distributions.

Theorem 3 The joint cdf of $\left(\boldsymbol{X}{[r]}, Y{(r)}\right)^{\top}$, for $t=\left(t_{1}^{\top}, t_{2}\right)^{\top} \in \mathbb{R}^{p+1}$, is given by
$$F_{[r],(r)}\left(t: \mu, \mathbf{\Sigma}, h^{(n p)}\right)-\sum_{\substack{i=1 \ j_{1}<\cdots}=\left(\xi_{i}^{}, \eta_{i j_{1}, \ldots, j_{r-1}}, \Omega_{i}^{}, \boldsymbol{\Gamma}{i j{1}, \ldots, j_{r-1}}, \boldsymbol{\Lambda}{i j{1}, \ldots, j_{r-1}}^{}\right)$$
with
\begin{aligned} \boldsymbol{\xi}{i}^{}=&\left(\begin{array}{l} \mu{\boldsymbol{X}{i}} \ \mu{Y_{i}} \end{array}\right), \quad \boldsymbol{\Omega}{i}^{}=\left(\begin{array}{cc} \boldsymbol{\Sigma}{\boldsymbol{X}{i}} \boldsymbol{X}{i} & \sigma_{\boldsymbol{X}{i} Y{i}} \ \boldsymbol{\sigma}{\boldsymbol{X}{i} Y_{i}} & \sigma_{Y_{i} Y_{i}} \end{array}\right) \ \boldsymbol{\Lambda}{i j{1}, \ldots, j_{r-1}}^{*}=&\left(\begin{array}{c} {\left[\mathbf{S}{i j{1}, \cdots, j_{r-1}}\left(\mathbf{1}{n-1} \boldsymbol{\sigma}{Y_{i} \boldsymbol{X}{i}}-\mathbf{\Sigma}{\boldsymbol{Y}{-i} \boldsymbol{X}{i}}\right)\right]^{\top}} \ {\left[\mathbf{S}{i j{1}, \cdots, j_{r-1}}\left(\mathbf{1}{n-1} \sigma{Y_{i} Y_{i}}-\boldsymbol{\sigma}{\boldsymbol{Y}{-i} Y_{i}}\right)\right]^{\top}} \end{array}\right) \end{aligned}
Proof This theorem can be proved by proceeding exactly as done in Theorem $2 .$

From Lemma 2, we can easily derive the conditional distribution of $\boldsymbol{X}{[r]}$, given $Y{(r)}$, as presented in the following corollary.

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

As an immediate consequence, we obtain the following corollary when $\left(\boldsymbol{X}{1}^{\top}, \ldots, \boldsymbol{X}{n}^{\top}\right)^{\top}$ follows a multivariate normal distribution.

Corollary 8 If $\left(\begin{array}{c}\mathbf{X}{1} \ \vdots \ \mathbf{X}{n}\end{array}\right) \sim N_{n p}(\boldsymbol{\mu}, \mathbf{\Sigma})$, then the conditional distribution of $\mathbf{X}{[r]}$, given $Y{(r)}=t_{2}$, for $t_{2} \in \mathbb{R}$, is given by
\begin{aligned} &F_{\mathbf{X}{[r]} \mid} \mid Y{(r)}=t_{2}\left(\mathbf{t}{1} ; \boldsymbol{\mu}, \boldsymbol{\Sigma}\right) \ &\quad=\sum{i=1}^{n} \sum_{\substack{j_{1}<\cdots<j_{r-1} \ 1 \leq j_{k} \leq n-1}} \pi_{i j_{1}, \cdots, j_{r-1}}^{n} G_{S U N_{p, n-1}}\left(\mathbf{t}{1} ; \theta{i j_{1}, \cdots, j_{r-1}}^{1,2}\left(t_{2}\right)\right), \mathbf{t}{1} \in \mathbb{R}^{p} \end{aligned} where the mixing probabilities are $$\pi{i j_{1}, \cdots, j_{r-1}}^{n}=\frac{\pi_{i j_{1}, \cdots, j_{r-1}} g \operatorname{SUN}{1, n-1}\left(t{2} ; \boldsymbol{\theta}{i j{1}, \cdots, j_{r-1}}\right)}{\sum_{\substack{i=1 \ j_{1}<\cdots<j_{r-1} \ 1 \leq j_{k} \leq n-1}} \sum_{i j_{1}, \cdots, j_{r-1}} g_{S U N_{1, n-1}}\left(t_{2} ; \boldsymbol{\theta}{i j{1}, \cdots, j_{r-1}}\right)}$$
and $\theta_{i j_{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right)$ and $\theta_{i j_{1}, \cdots, j_{r-1}}^{\prime}$ are as given in Corollary $7 .$
The conditional mean of $\mathbf{X}{[r]}$, given $Y{(r)}$ can be easily deduced, using the mean of $S U N_{n, m}(\theta)$ presented in Theorem 1 as $$E\left(\mathbf{X}{[r]} \mid Y{(r)}=t_{2}\right)=\sum_{i=1}^{n} \sum_{\substack{j_{1}<\cdots<<{l} \leq j{r-1} \ 1 \leq j_{k} \leq n-1}} \pi_{i j_{1}, \cdots, j_{r-1}}^{n} \mu_{i j_{1}, \cdots, j_{r-1}}\left(t_{2}\right)$$
where $\mu_{i j_{1}, \cdots, j_{r-1}}\left(t_{2}\right)$ denotes the mean of $S U N_{p, n-1}\left(\theta_{i j j_{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right)\right)$.

## 统计代写|生物统计代写biostatistics代考|Multivariate Student-t Case

Corollary 9 If $\left(\begin{array}{c}\mathbf{X}{1} \ \vdots \ \mathbf{X}{n}\end{array}\right) \sim t_{n p}(\mu, \Sigma, v)$, then the conditional distribution of $\mathbf{X}{[r]}$, given $Y{(r)}=t_{2}$ for $t_{2} \in \mathbb{R}$, is given by
$F_{\mathbf{X}{[r]} \mid Y{(r)}=t_{2}}\left(\mathbf{t}{1} ; \boldsymbol{\mu}, \mathbf{\Sigma}, v\right)$ $=\sum{i=1}^{n} \sum_{\substack{j_{1}<\cdots<j_{r-1} \ 1 \leq j_{k} \leq n-1}} \pi_{i j_{1}, \cdots, j_{r-1}}^{t} G_{S U T_{p, n-1}}\left(\mathbf{t}{1} ; \theta{i j_{1}, \cdots, j_{r-1}}^{\iota_{2}^{1.2}}\left(t_{2}\right), v+1\right), \mathbf{t}{1} \in \mathbb{R}^{p}$, where the mixing probabilities are $$\pi{i j_{1}, \cdots, j_{r-1}}^{t}=\frac{\pi_{i j_{1}, \cdots, j_{r-1}} g_{S U T_{1, n-1}}\left(t_{2} ; \boldsymbol{\theta}{i j{1}, \cdots, j_{r-1}^{\prime}}, v\right)}{\sum_{i=1}^{n} \sum_{\substack{j_{1}<\cdots<j_{1} \leq-1 \ 1 \leq j_{k} \leq n-1}} \pi_{i j j_{1}, \cdots, j_{r-1}} g_{S U T_{1, n-1}}\left(t_{2} ; \boldsymbol{\theta}{i j{1}, \cdots, j_{r-1}}^{\prime}, v\right)},$$
and
$$\theta_{i j_{1}, \cdots, j_{r-1}}^{* 1.2}\left(t_{2}\right)=\left(\xi_{i}^{1.2}\left(t_{2}\right), \eta_{i j_{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right), \boldsymbol{\Omega}{i}^{1.2}\left(t{2}\right), \boldsymbol{\Gamma}{i j{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right), \boldsymbol{\Lambda}{i j{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right)\right),$$
with $\boldsymbol{\xi}{i}^{1.2}\left(t{2}\right)$ and $\eta_{i j_{1}, \cdots, J_{r-1}}^{1.2}\left(t_{2}\right)$ being as in Corollary 7 , and
\begin{aligned} \boldsymbol{\Omega}{i}^{1.2}\left(t{2}\right) &=\boldsymbol{\Omega}{i}^{1.2} \frac{v+u{i}\left(t_{2}\right)}{v+1}, \ \boldsymbol{\Gamma}{i j{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right) &=\boldsymbol{\Gamma}{i j{1}, \cdots, j_{r-1}}^{1.2} \frac{v+u_{i}\left(t_{2}\right)}{v+1}, \ \boldsymbol{\Lambda}{i j{1}, \cdots, j_{r-1}}^{1.2}\left(t_{2}\right) &=\boldsymbol{\Lambda}{i j{1}, \cdots, j_{r-1}}^{1.2} \frac{v+u_{i}\left(t_{2}\right)}{v+1} . \end{aligned}
In a similar manner, the conditional mean of $\mathbf{X}{[r]}$, given $Y{(r)}$ is given by

$$E\left(\mathbf{X}{[r]} \mid Y{(r)}=t_{2}\right)=\sum_{\substack{i=1 \ j_{1}<\cdots<j_{r-1} \ 1 \leq j_{k} \leq n-1}}^{n} \pi_{i j_{1}, \cdots, j_{r-1}}^{t} \mu_{i j_{1}, \cdots, j_{r-1}}^{}\left(t_{2}\right)$$ where $\mu_{i j_{1}, \cdots, j_{r-1}^{}}\left(t_{2}\right)$ denotes the mean of $S U T_{p, n-1}\left(\theta_{i j_{1}, \cdots, j_{r-1}}{ }^{1.2}\left(t_{2}\right), v\right)$.

## 统计代写|生物统计代写biostatistics代考|Joint Distribution

$\$ \$$F_{-}\left{[r]\right.,(r)} }left(t: \mu, \mathbf{\Sigma}, h^{\wedge}{(n p)} \backslash \mathrm{~ I O m e g a _ { i } ^ { } , ~ \ b o l d s y m b o l { G a m m a } { i j { 1 } , ~ \ d o t s , ~ j _ { r – 1 } } , ~ \ b o l d s y m b o l {} with \boldsymbol{\xi} i=\left(\mu \boldsymbol{X} i \mu Y_{i}\right), \quad \boldsymbol{\Omega} i=\left(\boldsymbol{\Sigma} \boldsymbol{X}{i} \boldsymbol{X}{i} \quad \sigma_{\boldsymbol{X} i Y i} \boldsymbol{\sigma} \boldsymbol{X}{i} Y{i} \quad \sigma_{Y_{i} Y_{i}}\right) \boldsymbol{\Lambda} i j 1, \ldots, j_{r-1}{ }^{*}=\quad\left(\left[\mathbf{S} i j 1, \cdots, j_{r-1}\left(\mathbf{1} n-1 \boldsymbol{\sigma} Y_{i} \boldsymbol{X}{i}-\boldsymbol{\Sigma} \boldsymbol{Y}-i \boldsymbol{X}{i}\right)\right]^{\top}\left[\mathbf{S} i j 1, \cdots, j_{r-1}\left(\mathbf{1} n-1 \sigma Y_{i} Y_{i}-\boldsymbol{\sigma} \boldsymbol{Y}-i Y_{i}\right)\right]^{\top}\right) \ \$$

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

$$F_{\mathbf{X}[r] \mid} \mid Y(r)=t_{2}(\mathbf{t} 1 ; \boldsymbol{\mu}, \mathbf{\Sigma}) \quad=\sum i=1^{n} \sum_{j_{1} \cdots j_{r-1} 1 \leq j_{k} \leq n-1} \pi_{i j_{1} \cdots j_{r-1}}^{n} G_{S U N_{p n-1}}\left(\mathbf{t} 1 ; \theta i j_{1}, \cdots, j_{r-1} 1,2\left(t_{2}\right)\right), \mathbf{t} 1 \in \mathbb{R}^{p}$$

$$E\left(\mathbf{X}[r] \mid Y(r)=t_{2}\right)=\sum_{i=1}^{n} \sum_{j_{1}<\cdots<l \leq j r-11 \leq j_{k} \leq n-1} \pi_{i j j_{1}, \cdots, j_{r-1}}^{n} \mu_{i j_{1} \cdots, j_{r-1}}\left(t_{2}\right)$$

## 统计代写|生物统计代写biostatistics代考|Multivariate Student-t Case

$$\pi i j_{1}, \cdots, j_{r-1}{ }^{t}=\frac{\pi_{i j_{1}, \cdots, j_{r-1}} g_{S U T_{1, n-1}}\left(t_{2} ; \boldsymbol{\theta} i j 1, \cdots, j_{r-1}^{\prime}, v\right)}{\sum_{i=1}^{n} \sum_{j_{1}<\cdots<j_{1} \leq-11 \leq j_{k} \leq n-1} \pi_{i j j_{1}, \cdots, j r-1} g_{S U T 1, n-1}\left(t_{2} ; \boldsymbol{\theta} i j 1, \cdots, j_{r-1}^{\prime}, v\right)},$$

$$\theta_{i j_{1} \cdots j_{r-1}^{* 1}}^{* .2}\left(t_{2}\right)=\left(\xi_{i}^{1.2}\left(t_{2}\right), \eta_{i_{i}, \cdots j_{r-1}^{1.2}}\left(t_{2}\right), \boldsymbol{\Omega i}^{1.2}(t 2), \boldsymbol{\Gamma} i j 1, \cdots, j_{r-1}^{1.2}\left(t_{2}\right), \boldsymbol{\Lambda} i j 1, \cdots, j_{r-1}^{1.2}\left(t_{2}\right)\right),$$

$$\boldsymbol{\Omega} i^{1.2}(t 2)=\boldsymbol{\Omega} i^{1.2} \frac{v+u i\left(t_{2}\right)}{v+1}, \boldsymbol{\Gamma} i j 1, \cdots, j_{r-1}^{1.2}\left(t_{2}\right) \quad=\boldsymbol{\Gamma} i j 1, \cdots, j_{r-1}^{1.2} \frac{v+u_{i}\left(t_{2}\right)}{v+1}, \boldsymbol{\Lambda} i j 1, \cdots, j_{r-1}^{1.2}\left(t_{2}\right)=\boldsymbol{\Lambda} i j 1, \cdots, j_{r-1}{ }^{1.2} \frac{v+u_{i}\left(t_{2}\right)}{v+1} .$$

$$E\left(\mathbf{X}[r] \mid Y(r)=t_{2}\right)=\sum_{i=1}^{n} \sum_{j 1<\cdots<j_{r-1} 1 \leq j k \leq n-1} \pi_{i j 1, \cdots, j r-1}^{t} \mu_{i j j_{1}, \cdots, j r-1}\left(t_{2}\right)$$

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