统计代写|生物统计代写biostatistics代考|MPH701

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

统计代写|生物统计代写biostatistics代考|Extension to the Regression Case

We want to extend the methodology of Sect. $3.2$ to the regression setting where the location parameter varies across observations as a linear function of a set of $p$, say, explanatory variables, which are assumed to include the constant term, as it is commonly the case. If $x_{i}$ is the vector of covariates pertaining to the $i$ th subject, observation $y_{i}$ is now assumed to be drawn from ST $\left(\xi_{i}, \omega, \lambda, \nu\right)$ where
$$\xi_{i}=x_{i}^{\top} \beta, \quad i=1, \ldots, n,$$
for some $p$-dimensional vector $\beta$ of unknown parameters; hence now the parameter vector is $\theta=\left(\beta^{\top}, \omega, \lambda, v\right)^{\top}$. The assumption of independently drawn observations is retained.

The direct extension of the median as an estimate of location, which was used in Sect. 3.2, is an estimate of $\beta$ obtained by median regression, which corresponds to adoption of the least absolute deviations fitting criterion instead of the more familiar least squares. This can also be viewed as a special case of quantile regression, when the quantile level is set at $1 / 2$. A classical treatment of quantile regression

is Koenker (2005) and corresponding numerical work can be carried out using the $R$ package quantreg, see Koenker (2018), among other tools.

Use of median regression delivers an estimate $\tilde{\tilde{\beta}}^{m}$ of $\beta$ and a vector of residual values, $r_{i}=y_{i}-x_{i}^{\top} \tilde{\beta}^{m}$ for $i=1, \ldots, n$. Ignoring $\beta$ estimation errors, these residuals are values sampled from $\mathrm{ST}\left(-m_{0}, \omega^{2}, \lambda, v\right)$, where $m_{0}$ is a suitable value, examined shortly, which makes the distribution to have 0 median, since this is the target of the median regression criterion. We can then use the same procedure of Sect. 3.2, with the $y_{i}$ ‘s replaced the $r_{i}$ ‘s, to estimate $\omega, \lambda, v$, given that the value of $m_{0}$ is irrelevant at this stage.

The final step is a correction to the vector $\tilde{\beta}^{m}$ to adjust for the fact that $y_{i}-x_{i}^{\top} \beta$ should have median $m_{0}$, that is, the median of ST $(0, \omega, \lambda, v)$, not median 0 . This amounts to increase all residuals by a constant value $m_{0}$, and this step is accoomplishéd by sêtting a vectoor $\tilde{\beta}$ with all components equal tō $\tilde{\beta}^{m}$ except that the intercept term, $\beta_{0}$ say, is estimated by
$$\tilde{\beta}{0}=\tilde{\beta}{0}^{m}-\tilde{\omega} q_{2}^{\mathrm{ST}}$$
similarly to $(10)$

统计代写|生物统计代写biostatistics代考|Extension to the Multivariate Case

Consider now the case of $n$ independent observations from a multivariate $Y$ variable with density (6), hence $Y \sim \mathrm{ST}{d}(\xi, \Omega, \alpha, v)$. This case can be combined with the regression setting of Sect. 3.3, so that the $d$-dimensional location parameter varies for each observation according to $$\xi{i}^{\top}=x_{i}^{\top} \beta, \quad i=1, \ldots, n,$$
where now $\beta=\left(\beta_{\cdot 1}, \ldots, \beta_{\cdot d}\right)$ is a $p \times d$ matrix of parameters. Since we have assumed that the explanatory variables include a constant term, the regression case subsumes the one of identical distribution, when $p=1$. Hence we deal with the regression case directly, where the $i$ th observation is sampled from $Y_{i} \sim$ $\mathrm{ST}{d}\left(\xi{i}, \Omega, \alpha, v\right)$ and $\xi_{i}$ is given by (12), for $i=1, \ldots, n$.

Arrange the observed values in a $n \times d$ matrix $y=\left(y_{i j}\right)$. Application of the procedure presented in Sects. $3.2$ and $3.3$ separately to each column of $y$ delivers estimates of $d$ univariate models. Specifically, from the $j$ th column of $y$, we obtain estimates $\tilde{\theta}{j}$ and corresponding ‘normalized’ residuals $\tilde{z}{i j}$ :
$$\tilde{\theta}{j}=\left(\tilde{\beta}{\cdot j}^{\top}, \tilde{\omega}{j}, \tilde{\lambda}{j}, \tilde{v}{j}\right)^{\top}, \quad \tilde{z}{i j}=\tilde{\omega}{j}^{-1}\left(y{i j}-x_{i}^{\top} \tilde{\beta}_{\cdot j}\right)$$

where it must be recalled that the ‘normalization’ operation uses location and scale parameters, but these do not coincide with the mean and the standard deviation of the underlying random variable.

Since the meaning of expression (12) is to define a set of univariate regression modes with a common design matrix, the vectors $\tilde{\beta}{-1}, \ldots, \tilde{\beta}{\cdot d}$ can simply be arranged in a $p \times d$ matrix $\tilde{\beta}$ which represents an estimate of $\beta$.

The set of univariate estimates in (13) provide $d$ estimates for $v$, while only one such a value enters the specification of the multivariate ST distribution. We have adopted the median of $\tilde{v}{1}, \ldots, \tilde{v}{d}$ as the single required estimate, denoted $\tilde{v}$.

The scale quantities $\tilde{\omega}{1}, \ldots, \tilde{\omega}{d}$ estimate the square roots of the diagonal elements of $\Omega$, but off-diagonal elements require a separate estimation step. What is really required to estimate is the scale-free matrix $\bar{\Omega}$. This is the problem examined next.

If $\omega$ is the diagonal matrix formed by the squares roots of $\Omega_{11}, \ldots, \Omega_{\text {cld }}$, all variables $\omega^{-1}\left(Y_{i}-\xi_{i}\right)$ have distribution $\mathrm{ST}{d}(0, \bar{\Omega}, \alpha, v)$, for $i=1, \ldots, n$. Denote by $Z=\left(Z{1}, \ldots, Z_{d}\right)^{\top}$ the generic member of this set of variables. We are concerned with the distribution of the products $Z_{j} Z_{k}$, but for notational simplicity we focus on the specific product $W=Z_{1} Z_{2}$, since all other products are of similar nature.

We must then examine the distribution of $W=Z_{1} Z_{2}$ when $\left(Z_{1}, Z_{2}\right)$ is a bivariate ST variable. This looks at first to be a daunting task, but a major simplification is provided by consideration of the perturbation invariance property of symmetrymodulated distributions, of which the ST is an instance. For a precise exposition of this property, see for instance Proposition $1.4$ of Azzalini and Capitanio (2014), but in the present case it says that, since $W$ is an even function of $\left(Z_{1}, Z_{2}\right)$, its distribution does not depend on $\alpha$, and it coincides with the distribution of the case $\alpha=0$, that is, the case of a usual bivariate Student’s $t$ distribution, with dependence parameter $\bar{\Omega}_{12}$.

统计代写|生物统计代写biostatistics代考|Simulation Work to Compare Initialization Procedures

Several simulations runs have been performed to examine the performance of the proposed methodology. The computing environment was $\mathrm{R}$ version 3.6.0. The reference point for these evaluations is the methodology currently in use, as provided by the publicly available version of $R$ package $s n$ at the time of writing, namely version 1.5-4; see Azzalini (2019). This will be denoted ‘the current method’ in the following. Since the role of the proposed method is to initialize the numerical MLE search, not the initialization procedure per se, we compare the new and the current method with respect to final MLE outcome. However, since the numerical optimization method used after initialization is the same, any variations in the results originate from the different initialization procedures.

We stress again that in a vast number of cases the working of the current method is satisfactory and we are aiming at improvements when dealing with ‘awkward samples’. These commonly arise with ST distributions having low degrees of freedom, about $v=1$ or even less, but exceptions exist, such as the second sample in Fig. $2 .$

The primary aspect of interest is improvement in the quality of data fitting. This is typically expressed as an increase of the maximal achieved log-likelihood, in its penalized form. Another desirable effect is improvement in computing time.

The basic set-up for such numerical experiments is represented by simple random samples, obtained as independent and identically distributed values drawn from a named ST $(\xi, \omega, \lambda, v)$. In all cases we set $\xi=0$ and $\omega=1$. For the other ingredients, we have selected the following values:
$\lambda: 0, \quad 2, \quad 8$,
$v: 1,3,8$,
$n: 50,100,250,500$
and, for each combination of these values, $N=2000$ samples have been drawn.
The smallest examined sample size, $n=50$, must be regarded as a sort of ‘sensible lower bound’ for realistic fitting of flexible distributions such as the ST. In this respect, recall the cautionary note of Azzalini and Capitanio (2014, p. 63) about the fitting of a SN distribution with small sample sizes. Since the ST involves an additional parameter, notably one having a strong effect on tail behaviour, that annotation holds a fortiori here.

For each of the $3 \times 3 \times 4 \times 2000=72,000$ samples so generated, estimation of the parameters $(\xi, \omega, \lambda, \nu)$ has been carried out using the following methods.

统计代写|生物统计代写biostatistics代考|Extension to the Regression Case

X一世=X一世⊤b,一世=1,…,n,

b~0=b~0米−ω~q2小号吨

统计代写|生物统计代写biostatistics代考|Extension to the Multivariate Case

X一世⊤=X一世⊤b,一世=1,…,n,

θ~j=(b~⋅j⊤,ω~j,λ~j,在~j)⊤,和~一世j=ω~j−1(是一世j−X一世⊤b~⋅j)

（13）中的一组单变量估计提供d估计为在，而只有一个这样的值进入多元 ST 分布的规范。我们采用了 $\tilde{v} {1}、\ldots、\tilde{v} {d}的中位数一个s吨H和s一世nGl和r和q在一世r和d和s吨一世米一个吨和,d和n○吨和d\波浪号 {v}$。

λ:0,2,8,

n:50,100,250,500

有限元方法代写

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MATLAB代写

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