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A unified testing approach for log-symmetry using Fourier methods

Ganesh Vishnu Avhad, Sudheesh K. Kattumannil

TL;DR

This paper addresses the problem of goodness-of-fit testing for the class of log-symmetric distributions in positively skewed data. It introduces a Fourier-based, weighted $L^2$ test built on a characterization that $X_{(1)}$ and $1/X_{(n)}$ are identically distributed under log-symmetry, implementing the test via empirical characteristic functions. The authors establish the asymptotic distribution under the null and prove consistency under alternatives, then demonstrate favorable finite-sample performance through extensive simulations and real-data analyses, while emphasizing computational efficiency relative to existing methods. The work provides practical, fast tools for practitioners to assess log-symmetry and suggests directions for extensions to multivariate settings and regression contexts, with the tuning parameter $a$ offering a flexible sensitivity control.

Abstract

Continuous and strictly positive data that exhibit skewness and outliers frequently arise in many applied disciplines. Log-symmetric distributions provide a flexible framework for modeling such data. In this article, we develop new goodness-of-fit tests for log-symmetric distributions based on a recent characterization. These tests utilize the characteristic function as a novel tool and are constructed using an $L^2$-type weighted distance measure. The asymptotic properties of the resulting test statistic are studied. The finite-sample performance of the proposed method is assessed via Monte Carlo simulations and compared with existing procedures. The results under a range of alternative distributions indicate superior empirical power, while the proposed test also exhibits substantial computational efficiency compared to existing methods. The methodology is further illustrated using real data sets to demonstrate practical applicability.

A unified testing approach for log-symmetry using Fourier methods

TL;DR

This paper addresses the problem of goodness-of-fit testing for the class of log-symmetric distributions in positively skewed data. It introduces a Fourier-based, weighted test built on a characterization that and are identically distributed under log-symmetry, implementing the test via empirical characteristic functions. The authors establish the asymptotic distribution under the null and prove consistency under alternatives, then demonstrate favorable finite-sample performance through extensive simulations and real-data analyses, while emphasizing computational efficiency relative to existing methods. The work provides practical, fast tools for practitioners to assess log-symmetry and suggests directions for extensions to multivariate settings and regression contexts, with the tuning parameter offering a flexible sensitivity control.

Abstract

Continuous and strictly positive data that exhibit skewness and outliers frequently arise in many applied disciplines. Log-symmetric distributions provide a flexible framework for modeling such data. In this article, we develop new goodness-of-fit tests for log-symmetric distributions based on a recent characterization. These tests utilize the characteristic function as a novel tool and are constructed using an -type weighted distance measure. The asymptotic properties of the resulting test statistic are studied. The finite-sample performance of the proposed method is assessed via Monte Carlo simulations and compared with existing procedures. The results under a range of alternative distributions indicate superior empirical power, while the proposed test also exhibits substantial computational efficiency compared to existing methods. The methodology is further illustrated using real data sets to demonstrate practical applicability.
Paper Structure (11 sections, 4 theorems, 30 equations, 2 figures, 7 tables, 2 algorithms)

This paper contains 11 sections, 4 theorems, 30 equations, 2 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Test statistic based on a characterization
  3. Characterization of log-symmetry
  4. Test construction
  5. Asymptotic theory
  6. The limit distribution under the null hypothesis
  7. Consistency of the test
  8. Simulation study
  9. Real data analysis
  10. Concluding remarks
  11. Algorithms

Key Result

Theorem 1

Let $X, X_1, \dots, X_n$ be a random sample from a distribution $F_X$ with support on $\mathbb{R}^+$. For a fixed $i$ where $1 \leq i \leq n/2$ and $n > 1$, if the statistics $X_{(i)}$ and $1/X_{(n-i+1)}$ are identically distributed, then $X$ follows a log-symmetric distribution.

Figures (2)

  • Figure 1: Histogram, boxplot, and empirical distribution function of insulating fluid data set
  • Figure 2: Histogram, boxplot, and empirical distribution function of airborne communication transceiver data set

Theorems & Definitions (7)

  • Theorem 1: Ahmadi and Balakrishnan, 2024
  • Theorem 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Remark 1