In allometric studies, the joint distribution of the log-transformed morphometric variables is typically symmetric and with heavy tails. Moreover, in the bivariate case, it is customary to explain the morphometric variation of these variables by fitting a convenient line, as for example the first principal component (PC). To account for all these peculiarities, we propose the use of multiple scaled symmetric (MSS) distributions. These distributions have the advantage to be directly defined in the PC space, the kind of symmetry involved is less restrictive than the commonly considered elliptical symmetry, the behavior of the tails can vary across PCs, and their first PC is less sensitive to outliers. In the family of MSS distributions, we also propose the multiple scaled shifted exponential normal distribution, equivalent of the multivariate shifted exponential normal distribution in the MSS framework. For the sake of parsimony, we also allow the parameter governing the leptokurtosis on each PC, in the considered MSS distributions, to be tied across PCs. From an inferential point of view, we describe an EM algorithm to estimate the parameters by maximum likelihood, we illustrate how to compute standard errors of the obtained estimates, and we give statistical tests and confidence intervals for the parameters. We use artificial and real allometric data to appreciate the advantages of the MSS distributions over well-known elliptically symmetric distributions and to compare the robustness of the line from our models with respect to the lines fitted by well-established robust and non-robust methods available in the literature.

中文翻译：

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异速生长研究中的多尺度对称分布

在异速生长研究中，对数变换的形态测量变量的联合分布通常是对称的并且具有重尾。此外，在双变量情况下，习惯上通过拟合一条方便的线来解释这些变量的形态变化，例如第一主成分 (PC)。为了解释所有这些特性，我们建议使用多个比例对称 (MSS) 分布。这些分布的优点是可以直接在 PC 空间中定义，所涉及的对称类型比通常认为的椭圆对称限制更少，尾部的行为可以在 PC 之间变化，并且它们的第一个 PC 对异常值不太敏感。在 MSS 分布族中，我们还提出了多尺度移位指数正态分布，等效于 MSS 框架中的多元移位指数正态分布。为了简洁起见，我们还允许在所考虑的 MSS 分布中控制每台 PC 上的尖峰态的参数在 PC 之间绑定。从推理的角度来看，我们描述了一种通过最大似然估计参数的 EM 算法，我们说明了如何计算所获得估计值的标准误差，并给出了参数的统计检验和置信区间。我们使用人工和真实的异速生长数据来了解 MSS 分布相对于众所周知的椭圆对称分布的优势，并比较我们模型中线的稳健性与通过成熟的稳健和非稳健方法拟合的线在文献中。