Allometric analysis using the multivariate shifted exponential normal distribution.

Area of Science:

• Statistics
• Biometry
• Allometry

Background:

• Allometric studies often involve log-transformed morphometric variables with elliptical and heavy-tailed distributions.
• Existing multivariate normal distributions may not adequately capture these data characteristics.

Purpose of the Study:

• Introduce the multivariate shifted exponential normal (MSEN) distribution.
• Provide a heavy-tailed, elliptical generalization of the multivariate normal (MN) distribution for allometric data analysis.

Main Methods:

• The MSEN distribution is defined as a multivariate normal scale mixture (MNSM) using a shifted exponential mixing distribution.
• Maximum likelihood (ML) estimates are computed using the expectation-maximization (EM) algorithm.
• The tail weight parameter is estimated from data, enabling robust mean vector estimation.

Main Results:

• The MSEN distribution has a simple closed-form probability density function with one additional parameter for tail weight control.
• The first four moments exist, and excess kurtosis can be any positive value.
• The EM algorithm provides computationally simplified, closed-form updates for parameters.
• Robust estimates of the mean vector are automatically obtained through data-driven tail weight estimation.

Conclusions:

• The MSEN distribution effectively models multivariate allometric data with heavy tails.
• It offers a useful alternative to existing elliptical distributions for morphometric analyses.
• The method provides robust parameter estimation, enhancing the reliability of allometric studies.