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.

翻译：近年来，用于建模角数据的灵活分布受到了广泛关注，持续的研究工作将现有的圆形模型进行扩展，以在捕捉多样化的角行为方面提供更大的灵活性。本文介绍并研究了w3PL分布，这是一种通过为包裹林德利分布引入两个额外的形状参数而得到的圆形模型。所提出的推广在保持解析易处理性并包含现有圆形模型作为特例的同时，增强了在建模集中度和偏度方面的灵活性。推导了概率密度函数、累积分布函数和三角矩的闭式表达式，从而允许通过解析方法研究关键的分布性质。对新提出的圆形模型的分布模态进行了表征，并研究了其不变性的本质。参数估计在正则化最大似然框架内展开，模拟研究证明了可靠的参数恢复能力和稳定的有限样本性能。对来自地质学、海洋生物学和金融学的角数据集的实证应用，说明了该模型的实际意义，并显示出相对于现有圆形替代模型的拟合优度提升。