The Wrapped Q-Lindley Distribution: A New Circular Model with Statistical Properties, Classical and Bayesian Inference, and Applications to Directional Data

Circular data arise naturally in a wide range of scientific fields, including biology, geology, meteorology, and environmental sciences, where conventional linear statistical methods are inappropriate unless the circular structure of the observations is taken into account. Motivated by the need for parsimonious, analytically tractable models for unimodal directional data, we introduce a new one-parameter circular model, the Wrapped Q-Lindley (WQL) distribution. The proposed model is obtained by wrapping the Q-Lindley distribution and has a specific unimodal asymmetric shape. Closed-form or analytically tractable expressions are derived for several important distributional properties, including the characteristic function, trigonometric moments, circular descriptive measures, and Rényi entropy. We investigate parameter estimation using ten classical estimation methods, along with Bayesian inference, under four symmetric and asymmetric loss functions. A Monte Carlo simulation study evaluates and compares the finite-sample performance of the competing estimators. The practical usefulness of the proposed model is illustrated through two real directional datasets involving sun compass orientation measurements of starhead topminnows and wind direction observations. Comparisons with several established circular models indicate that the WQL distribution provides competitive fitting performance for the considered datasets. We discuss its parsimonious structure and analytical tractability, along with the implications of its structural discontinuity at the wrap point, in the context of directional data analysis.