Paper: arXiv 2311.18176

Authors: Baishuai Zuo, Narayanaswamy Balakrishnan, Chuancun Yin

Abstract

This paper examines eight measures of skewness and Mardia measure of kurtosis for skew-elliptical distributions. Multivariate measures of skewness considered include Mardia, Malkovich-Afifi, Isogai, Song, Balakrishnan-Brito-Quiroz, M$\acute{o}$ri, Rohatgi and Sz$\acute{e}$kely, Kollo and Srivastava measures. We first study the canonical form of skew-elliptical distributions, and then derive exact expressions of all measures of skewness and kurtosis for the family of skew-elliptical distributions, except for Song’s measure. Specifically, the formulas of these measures for skew normal, skew $t$, skew logistic, skew Laplace, skew Pearson type II and skew Pearson type VII distributions are obtained. Next, as in Malkovich and Afifi (1973), test statistics based on a random sample are constructed for illustrating the usefulness of the established results. In a Monte Carlo simulation study, different measures of skewness and kurtosis for $2$-dimensional skewed distributions are calculated and compared. Finally, real data is analyzed to demonstrate all the results.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 6.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical treatment by deriving exact closed-form expressions for multiple multivariate skewness and kurtosis measures across the skew-elliptical family. Empirical components include Monte Carlo simulations and real-data applications, though it remains predominantly theoretical-statistical research.

Research Flowchart

  flowchart TD
    A[Goal: Analyze multivariate skewness & kurtosis for skew-elliptical distributions] --> B[Derive Canonical Form of Skew-Elliptical Distributions]
    B --> C[Derive Exact Formulas for 8 Skewness Measures & Mardia Kurtosis]
    C --> D[Inputs: Skew Normal, Skew t, Logistic, Laplace & Pearson Types II/VII]
    D --> E[Construct Sample Test Statistics & Perform Monte Carlo Simulations]
    E --> F[Apply Models & Measures to Real-World Quantitative Data]
    F --> G[Outcomes: Validated Theoretical Expressions & Comparative Performance Metrics]

Cited References (8)

  1. Normal linear models with lattice conditional independence restrictions (1994) — link
  2. A graphical characterization of lattice conditional independence models (1997)
  3. General Lattice Theory (2003)
  4. Graphical Models (1996)
  5. Totally ordered multivariate linear models (1993)
  6. Lattice models for conditional independence in a multivariate normal distribution (1993)
  7. Iterative conditional fitting for discrete chain graph models (2008)
  8. On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs (1995)

0 of 8 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.