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A flexible wrapped Lindley-type distribution for angular data modelling

Johan Ferreira, Delene van Wyk-de Ridder, Janet van Niekerk

Abstract

Flexible distributions for modelling angular data have received considerable attention in recent years, with ongoing work extending existing circular models to provide greater flexibility in capturing diverse angular behaviours. In this paper, we introduce and study the w3PL distribution, a circular model obtained by extending the wrapped Lindley distribution by incorporating two additional shape parameters. The proposed generalisation increases flexibility in modelling concentration and skewness while preserving analytical tractability and encompassing existing circular models as special cases. Closed-form expressions for the probability density function, cumulative distribution function, and trigonometric moments are derived, allowing key distributional properties to be studied analytically. The distributional modality is characterised, and the nature of invariance is investigated for the newly proposed circular model. Parameter estimation is developed within a regularised maximum likelihood framework, and a simulation study demonstrates reliable parameter recovery and stable finite-sample performance. Applications to angular datasets from geology, marine biology, and finance illustrate the model's practical significance and show improved fit relative to existing circular alternatives.

A flexible wrapped Lindley-type distribution for angular data modelling

Abstract

Flexible distributions for modelling angular data have received considerable attention in recent years, with ongoing work extending existing circular models to provide greater flexibility in capturing diverse angular behaviours. In this paper, we introduce and study the w3PL distribution, a circular model obtained by extending the wrapped Lindley distribution by incorporating two additional shape parameters. The proposed generalisation increases flexibility in modelling concentration and skewness while preserving analytical tractability and encompassing existing circular models as special cases. Closed-form expressions for the probability density function, cumulative distribution function, and trigonometric moments are derived, allowing key distributional properties to be studied analytically. The distributional modality is characterised, and the nature of invariance is investigated for the newly proposed circular model. Parameter estimation is developed within a regularised maximum likelihood framework, and a simulation study demonstrates reliable parameter recovery and stable finite-sample performance. Applications to angular datasets from geology, marine biology, and finance illustrate the model's practical significance and show improved fit relative to existing circular alternatives.
Paper Structure (18 sections, 5 theorems, 64 equations, 7 figures, 8 tables)

This paper contains 18 sections, 5 theorems, 64 equations, 7 figures, 8 tables.

Table of Contents

  1. Introduction
  2. A wrapped 3PL distribution
  3. Trigonometric moments
  4. Modality of the w3PL
  5. The property of invariance
  6. Estimation and sampling
  7. Estimation using maximum likelihood
  8. Sampling and simulation design
  9. Numerical examples
  10. Simulation studies
  11. Real data
  12. Cross-bed measurements
  13. Travel directions of blue periwinkles
  14. Credit-card transaction timestamp data
  15. Conclusion
  16. ...and 3 more sections

Key Result

Theorem 2.1

Suppose $X$ follows the 3PL distribution with the PDF eq:3PLindley PDF with parameters $(\kappa,\alpha,\beta)$. Then, the corresponding circular w3PL variable $\Theta$ has the following PDF: where $\theta \in [0,2\pi)$ and $\kappa,\beta>0$ and $\alpha\kappa+\beta>0$. Where relevant, this distribution will be denoted by w3PL. We note here that, as in the linear case, the "usual" wrapped Lindley mo

Figures (7)

  • Figure 1: w3PL PDF and CDF plots for different parameter configurations
  • Figure 2: Summary measures of the w3PL distribution as functions of $\kappa$ for selected $(\alpha,\beta)$ configurations
  • Figure 3: Interior-mode regime when $0<\kappa<\kappa_{r}$
  • Figure 4: Boundary-mode regime when $\kappa\geq\kappa_{r}$
  • Figure 5: Model fits to the cross-bed measurement data
  • ...and 2 more figures

Theorems & Definitions (10)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.1
  • Theorem 2.3
  • Remark 2.1
  • Theorem 2.4
  • proof : Proof of Theorem \ref{['pr:cWrapped 3PL PDF']}
  • proof : Proof of Theorem \ref{['pr:cWrapped 3PL CDF']}
  • proof : Proof of Lemma \ref{['lem:3PL characteristic function']}
  • proof : Proof of Theorem \ref{['pr: Mode of cWrapped 3PL']}