Abstract
The Cardioid (C) distribution is one of the most important models for modeling circular data. Although some of its structural properties have been derived, this distribution is not appropriate for asymmetry and multimodal phenomena in the circle, and then extensions are required. There are various general methods that can be used to produce circular distributions. This paper proposes four extensions of the C distribution based on the beta, Kumaraswamy, gamma, and Marshall–Olkin generators. We obtain a unique linear representation of their densities and some mathematical properties. Inference procedures for the parameters are also investigated. We perform two applications on real data, where the new models are compared to the C distribution and one of its extensions.
Highlights
Fitting densities to data has a long history
We propose four new circular distributions called the beta Cardioid, Kumaraswamy Cardioid (KwC), gamma
2π π θ )φ for x ∈ R \ {2πk : k ∈ Z}. This case is denoted by X ∼ KwC (θ, φ, μ, ρ)
Summary
Fitting densities to data has a long history. Statistical distributions are very useful in describing and predicting real world phenomena. Recent developments address definitions of new families that extend well-known distributions and, at the same time, provide great flexibility in modeling real data. As one of the most used circular distributions, the two-parameter Cardioid (C) law was pioneered by Jeffreys [15] for describing directional spectra of ocean waves This model has a cumulative distribution function (cdf), G ( x ) = G ( x; μ, ρ), and probability density function (pdf), g( x ) = g( x; μ, ρ), given by (for 0 < x ≤ 2π). Based on mixtures of one-dimensional Langevin distributions, Qiu and Wu [22] derived a new information criterion to cluster circular data. We understand that these works motivate our proposals as the potential inputs for future clustering structures.
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