统计代写|生物统计代写biostatistics代考|BIOL 220

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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|Flexible Distributions: The Skew-t Case

In the context of distribution theory, a central theme is the study of flexible parametric families of probability distributions, that is, families allowing substantial variation of their behaviour when the parameters span their admissible range.

For brevity, we shall refer to this domain with the phrase ‘flexible distributions’. The archetypal construction of this logic is represented by the Pearson system of curves for univariate continuous variables. In this formulation, the density function is regulated by four parameters, allowing wide variation of the measures of skewness and kurtosis, hence providing much more flexibility than in the basic case represented by the normal distribution, where only location and scale can be adjusted.

Since Pearson times, flexible distributions have remained a persistent theme of interest in the literature, with a particularly intense activity in recent years. A prominen feature of newer developments is the increased sonsideration for multivariate distributions, reflecting the current availability in applied work of larger datasets, both in sample size and in dimensionality. In the multivariate setting, the various formulations often feature four blocks of parameters to regulate location, scale, skewness and kurtosis.

While providing powerful tools for data fitting, flexible distributions also pose some challenges when we enter the concrete estimation stage. We shall be working with maximum likelihood estimation (MLE) or variants of it, but qualitatively similar issues exist for other criteria. Explicit expressions of the estimates are out of the question; some numerical optimization procedure is always involved and this process is not so trivial because of the larger number of parameters involved, as compared with fitting simpler parametric models, such as a Gamma or a Beta distribution. Furthermore, in some circumstances, the very flexibility of these parametric families can lead to difficulties: if the data pattern does not aim steadily towards a certain point of the parameter space, there could be two or more such points which constitute comparably valid candidates in terms of log-likelihood or some other estimation criterion. Clearly, these problems are more challenging with small sample size, later denoted $n$, since the log-likelihood function (possibly tuned by a prior distribution) is relatively more flat, but numerical experience has shown that they can persist even for fairly large $n$, in certain cases.

统计代写|生物统计代写biostatistics代考|The Skew-t Distribution: Basic Facts

Before entering our actual development, we recall some basic facts about the ST parametric family of continuous distributions. In its simplest description, it is obtained as a perturbation of the classical Student’s $t$ distribution. For a more specific description, start from the univariate setting, where the components of the family are identified by four parameters. Of these four parameters, the one denoted $\xi$ in the following regulates the location of the distribution; scale is regulated by the positive parameter $\omega$; shape (representing departure from symmetry) is regulated by $\lambda$; tail-weight is regulated by $v$ (with $v>0$ ), denoted ‘degrees of freedom’ like for a classical $t$ distribution.

It is convenient to introduce the distribution in the ‘standard case’, that is, with location $\xi=0$ and scale $\omega=1$. In this case, the density function is
$$t(z ; \lambda, v)=2 t(z ; v) T\left(\lambda z \sqrt{\frac{v+1}{v+z^{2}}} ; v+1\right), \quad z \in \mathbb{R}$$

where
$$t(z ; v)=\frac{\Gamma\left(\frac{1}{2}(v+1)\right)}{\sqrt{\pi v} \Gamma\left(\frac{1}{2} v\right)}\left(1+\frac{z^{2}}{v}\right)^{-(v+1) / 2}, \quad z \in \mathbb{R}$$
is the density function of the classical Student’s $t$ on $v$ degrees of freedom and $T(\cdot ; v)$ denotes its distribution function; note however that in (1) this is evaluated with $v+1$ degrees of freedom. Also, note that the symbol $t$ is used for both densities in (1) and (2), which are distinguished by the presence of either one or two parameters.

If $Z$ is a random variable with density function (1), the location and scale transform $Y=\xi+\omega Z$ has density function
$$t_{Y}(x ; \theta)=\omega^{-1} t(z ; \lambda, v), \quad z=\omega^{-1}(x-\xi),$$
where $\theta=(\xi, \omega, \lambda, v)$. In this case, we write $Y \sim \operatorname{ST}\left(\xi, \omega^{2}, \lambda, v\right)$, where $\omega$ is squared for similarity with the usual notation for normal distributions.

When $\lambda=0$, we recover the scale-and-location family generated by the $t$ distribution (2). When $v \rightarrow \infty$, we obtain the skew-normal (SN) distribution with parameters $(\xi, \omega, \lambda)$, which is described for instance by Azzalini and Capitanio (2014, Chap. 2). When $\lambda=0$ and $v \rightarrow \infty$, (3) converges to the $\mathrm{N}\left(\xi, \omega^{2}\right)$ distribution.

Some instances of density (1) are displayed in the left panel of Fig. 1. If $\lambda$ was replaced by $-\lambda$, the densities would be reflected on the opposite side of the vertical axis, since $-Y \sim \operatorname{ST}\left(-\xi, \omega^{2},-\lambda, \nu\right)$.

统计代写|生物统计代写biostatistics代考|Basic General Aspects

The high flexibility of the ST distribution makes it particularly appealing in a wide range of data fitting problems, more than its companion, the SN distribution. Reliable techniques for implementing connected MLE or other estimation methods are therefore crucial.

From the inference viewpoint, another advantage of the ST over the related SN distribution is the lack of a stationary point at $\lambda=0$ (or $\alpha=0$ in the multivariate case), and the implied singularity of the information matrix. This stationary point of the SN is systematic: it occurs for all samples, no matter what $n$ is. This peculiar aspect has been emphasized more than necessary in the literature, considering that it pertains to a single although important value of the parameter. Anyway, no such problem exists under the ST assumption. The lack of a stationary point at the origin was first observed empirically and welcomed as ‘a pleasant surprise’ by Azzalini and Capitanio (2003), but no theoretical explanation was given. Additional numerical evidence in this direction has been provided by Azzalini and Genton (2008). The theoretical explanation of why the SN and the ST likelihood functions behave differently was finally established by Hallin and Ley (2012).

Another peculiar aspect of the SN likelihood function is the possibility that the maximum of the likelihood function occurs at $\lambda=\pm \infty$, or at $|\alpha| \rightarrow \infty$ in the multivariate case. Note that this happens without divergence of the likelihood function, but only with divergence of the parameter achieving the maximum. In this respect the SN and the ST model are similar: both of them can lead to this pattern.
Differently from the stationarity point at the origin, the phenomenon of divergent estimates is transient: it occurs mostly with small $n$, and the probability of its occurrence decreases very rapidly when $n$ increases. However, when it occurs for the $n$ available data, we must handle it. There are different views among statisticians on whether such divergent values must be retained as valid estimates or they must be rejected as unacceptable. We embrace the latter view, for the reasons put forward by Azzalini and Arellano-Valle (2013), and adopt the maximum penalized likelihood estimate (MPLE) proposed there to prevent the problem. While the motivation for MPLE is primarily for small to moderate $n$, we use it throughout for consistency.
There is an additional peculiar feature of the ST log-likelihood function, which however we mention only for completeness, rather than for its real relevance. In cases when $v$ is allowed to span the whole positive half-line, poles of the likelihood function must exist near $v=0$, similarly to the case of a Student’s $t$ with unspecified degrees of freedom. This problem has been explored numerically by Azzalini and Capitanio (2003, pp. 384-385), and the indication was that these poles must exist at very small values of $v$, such as $\hat{v}=0.06$ in one specific instance.

This phenomenon is qualitatively similar to the problem of poles of the likelihood function for a finite mixture of continuous distributions. Even in the simple case of univariate normal components, there always exist $n$ poles on the boundary of the parameter space if the standard deviations of the components are unrestricted; see for instance Day (1969, Section 7). The problem is conceptually interesting, in both settings, but in practice it is easily dealt with in various ways. In the ST setting, the simplest solution is to impose a constraint $v>v_{0}>0$ where $v_{0}$ is some very small value, such as $v_{0}=0.1$ or $0.2$. Even if fitted to data, a $t$ or ST density with $v<0.1$ would be an object hard to use in practice.

统计代写|生物统计代写biostatistics代考|Basic General Aspects

ST 分布的高度灵活性使其在广泛的数据拟合问题中特别有吸引力，超过了它的同伴 SN 分布。因此，实现互联 MLE 或其他估计方法的可靠技术至关重要。

SN 似然函数的另一个特殊方面是似然函数的最大值出现在λ=±∞， 或|一个|→∞在多变量情况下。请注意，这种情况在似然函数没有发散的情况下发生，但只有在参数的发散达到最大值的情况下才会发生。在这方面，SN 和 ST 模型是相似的：它们都可以导致这种模式。

ST 对数似然函数还有一个额外的特殊功能，但是我们仅出于完整性而不是其真正相关性而提及它。在某些情况下在允许跨越整个正半线，似然函数的极点必须存在于附近在=0，类似于学生的情况吨具有未指定的自由度。Azzalini 和 Capitanio (2003, pp. 384-385) 对这个问题进行了数值研究，表明这些极点必须以非常小的值存在在， 如在^=0.06在一个特定的情况下。

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