### 统计代写|生物统计代写biostatistics代考|STA 310

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|Numerical Aspects and Some Illustrations

Since, on the computational side, we shall base our work the R package sn, described by Azzalini (2019), it is appropriate to describe some key aspects of this package. There exists a comprehensive function for model fitting, called selm, but the actual numerical work in case of an ST model is performed by functions st. mple and mst. mple, in the univariate and the multivariate case, respectively. To numerical efficiency, we shall be using these functions directly, rather than via selm. As their names suggest, st. mple and mst. mple perform MPLE, but they can be used for classical MLE as well, just by omitting the penalty function. The rest of the description refers to st. mple, but mst. mple follows a similar scheme.
In the univariate case, denote by $\theta=(\xi, \omega, \alpha, \nu)^{\top}$ the parameters to be cstimatcd, or possibly $\theta=\left(\beta^{\top}, w, \alpha, v\right)^{\top}$ when a lincar regrcssion mudel is introduced for the location parameter, in which case $\beta$ is a vector of $p$ regression coefficients. Denote by $\log L(\theta)$ the log-likelihood function at point $\theta$. If no starting values are supplied, the first operation of st.mple is to fit a linear model to the available explanatory variables; this reduces to the constant covariate value 1 if $p=1$. For the residuals from this linear fit, sample cumulants of order up to four are computed, hence including the sample variance. An inversion from these

values to $\theta$ may or may not be possible, depending on whether the third and fourth sample cumulants fall in the feasible region for the ST family. If the inversion is successful, initial values of the parameters are so obtained; if not, the final two components of $\theta$ are set at $(\alpha, v)=(0,10)$, retaining the other components from the linear fit. Starting from this point, MLE or MPLE is searched for using a general numerical optimization procedure. The default procedure for performing this step is the $\mathrm{R}$ function nlminb, supplied with the score functions besides the log-likelihood function. We shall refer, comprehensively, to this currently standard procedure as ‘method M0’.

In all our numerical work, method M0 uses st. mple, and the involved function nlminb, with all tuning parameters kept at their default values. The only activated option is the one switching between MPLE and MLE, and even this only for the work of the present section. Later on, we shall always use MPLE, with penalty function Openalty which implements the method proposed in Azzalini and Arellano-Valle (2013).

We start our numerical work with some illustrations, essentially in graphical form, of the log-likelihood generated by some simulated datasets. The aim is to provide a direct perception, although inevitably limited, of the possible behaviour of the log-likelihood and the ensuing problems which it poses for MLE search and other inferential procedures. Given this aim, we focus on cases which are unusual, in some way or another, rather than on ‘plain cases’.

The type of graphical display which we adopt is based on the profile loglikelihood function of $(\alpha, v)$, denoted $\log L_{p}(\alpha, v)$. This is obtained, for any given $(\alpha, v)$, by maximizing $\log L(\theta)$ with respect to the remaining parameters. To simplify readability, we transform $\log L_{p}(\alpha, v)$ to the likelihood ratio test statistic, also called ‘deviance function’:
$$D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}$$
where $\log L_{p}(\hat{\alpha}, \hat{v})$ is the overall maximum value of the log-likelihood, equivalent to $\log L(\hat{\theta})$. The concept of deviance applies equally to the penalized log-likelihood.
The plots in Fig. 2 displays, in the form of contour level plots, the behaviour of $D(\alpha, v)$ for two artificially generated samples, with $v$ expressed on the logarithmic scale for more convenient readability. Specifically, the top plots refer to a sample of size $n=50$ drawn from the $\operatorname{ST}(0,1,1,2)$; the left plot, refers to the regular log-likelihood, while the right plot refers to the penalized log-likelihood. The plots include marks for points of special interest, as follows:
$\Delta$ the true parameter point;
o the point having maximal (penalized) log-likelihood on a $51 \times 51$ grid of points spanning the plotted area;

• the MLE or MPLE point selected by method M0;
• the preliminary estimate to be introduced in Sect. 3.2, later denoted M1;
$\times$ the MLE or MPLE point selected by method M2 presented later in the text.

## 统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

We have seen in Sect. 2 the ST log-likelihood function can be problematic; it is then advisable to select carefully the starting point for the MLE search. While contrasting the risk of landing on a local maximum, a connected aspect of interest is to reduce the overall computing time. Here are some preliminary considerations about the stated target.

Since these initial estimates will be refined by a subsequent step of log-likelihood maximization, there is no point in aiming at a very sophisticate method. In addition, we want to keep the involved computing header as light as possible. Therefore, we want a method which is simple and quick to compute; at the same time, it should be reasonably reliable, hopefully avoiding nonsensical outcomes.

Another consideration is that we cannot work with the methods of moments, or some variant of it, as this would impose a condition $v>4$, bearing in mind the constraints recalled in Sect. 1.2. Since some of the most interesting applications of ST-based models deal with very heavy tails, hence with low degrees of freedom, the condition $v>4$ would be unacceptable in many important applications. The implication is that we have to work with quantiles and derived quantities.

To ease exposition, we begin by presenting the logic in the basic case of independent observations from a common univariate distribution $\mathrm{ST}\left(\xi, \omega^{2}, \lambda, v\right)$. The first step is to select suitable quantile-based measures of location, scale,

asymmetry and tail-weight. The following list presents a set of reasonable choices; these measures can be equally referred to a probability distribution or to a sample, depending on the interpretation of the terms quantile, quartile and alike.

Location The median is the obvious choice here; denote it by $q_{2}$, since it coincides with the second quartile.

Scale A commonly used measure of scale is the semi-interquartile difference, also called quartile deviation, that is
$$d_{q}=\frac{1}{2}\left(q_{3}-q_{1}\right)$$
where $q_{j}$ denotes the $j$ th quartile; see for instance Kotz et al. (2006, vol. 10, p. 6743).

Asymmetry A classical non-parametric measure of asymmetry is the so-called Bowley’s measure
$$G=\frac{\left(q_{3}-q_{2}\right)-\left(q_{2}-q_{1}\right)}{q_{3}-q_{1}}=\frac{q_{3}-2 q_{2}+q_{1}}{2 d_{q}}$$
see Kotz et al. (2006, vol. 12, p. 7771-3). Since the same quantity, up to an inessential difference, had previously been used by Galton, some authors attribute to him its introduction. We shall refer to $G$ as the Galton-Bowley measure.

Kurtosis A relatively more recent proposal is the Moors measure of kurtosis, presented in Moors (1988),
$$M=\frac{\left(e_{7}-e_{5}\right)+\left(e_{3}-e_{1}\right)}{e_{6}-e_{2}}$$
where $e_{j}$ denotes the $j$ th octile, for $j=1, \ldots, 7$. Clearly, $e_{2 j}=q_{j}$ for $j=$ $1,2,3$

## 统计代写|生物统计代写biostatistics代考|Inversion of Quantile-Based Measures to ST Parameters

For the inversion of the parameter set $Q=\left(q_{2}, d_{q}, G, M\right)$ to $\theta=(\xi, \omega, \lambda, v)$, the first stage considers only the components $(G, M)$ which are to be mapped to $(\lambda, v)$, exploiting the invariance of $G$ and $M$ with respect to location and scale. Hence, at this stage, we can work assuming that $\xi=0$ and $\omega=1$.

Start by computing, for any given pair $(\lambda, v)$, the set of octiles $e_{1}, \ldots, e_{7}$ of $\mathrm{ST}(0,1, \lambda, v)$, and from here the corresponding $(G, M)$ values. Operationally, we have computed the ST quantiles using routine qst of package sn. Only nonnegative values of $\lambda$ need to be considered, because a reversal of the $\lambda$ sign simply reverses the sign of $G$, while $M$ is unaffected, thanks to the mirroring property of the ST quantiles when $\lambda$ is changed to $-\lambda$.

Initially, our numerical exploration of the inversion process examined the contour level plots of $G$ and $M$ as functions of $\lambda$ and $v$, as this appeared to be the more natural approach. Unfortunately, these plots turned out not to be useful, because of the lack of a sufficiently regular pattern of the contour curves. Therefore these plots are not even displayed here.

A more useful display is the one adopted in Fig. 3, where the coordinate axes are now $G$ and $M$. The shaded area, which is the same in both panels, represents the set of feasible $(G, M)$ points for the ST family. In the first plot, each of the black lines indicates the locus of points with constant values of $\delta$, defined by (4), when $v$ spans the positive half-line; the selected $\delta$ values are printed at the top of the shaded area, when feasible without clutter of the labels. The use of $\delta$ instead of $\lambda$ simply yields a better spread of the contour lines with different parameter values, but it is conceptually irrelevant. The second plot of Fig. 3 displays the same admissible region with superimposed a different type of loci, namely those corresponding to specified values of $v$, when $\delta$ spans the $[0,1]$ interval; the selected $v$ values are printed on the left side of the shaded area.

Details of the numerical calculations are as follows. The Galton-Bowley and the Moors measures have been evaluated over a $13 \times 25$ grid of points identified by the selected values
\begin{aligned} \delta^{}=&(0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,0.95,0.99,1) \ v^{}=&(0.30,0.32,0.35,0.40,0.45,0.50,0.60,0.70,0.80,0.90,1,1.5,2\ &3,4,5,7,10,15,20,30,40,50,100, \infty) \end{aligned}

## 统计代写|生物统计代写biostatistics代考|Numerical Aspects and Some Illustrations

D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}D(\alpha, v)=2\left{\log L_{p}(\hat{\alpha}, \hat{v})-\log L_{p}(\alpha, v)\right}

Δ真正的参数点；
o 在 a 上具有最大（惩罚）对数似然的点51×51跨越绘图区域的点网格；

• 方法 M0 选择的 MLE 或 MPLE 点；
• 将在 Sect 中介绍的初步估计。3.2，后面记为M1；
×文中稍后介绍的方法 M2 选择的 MLE 或 MPLE 点。

## 统计代写|生物统计代写biostatistics代考|Preliminary Remarks and the Basic Scheme

dq=12(q3−q1)

G=(q3−q2)−(q2−q1)q3−q1=q3−2q2+q12dq

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