A tractable and interpretable four-parameter family of unimodal distributions on the circle

On optimal tests for circular reflective symmetry about an unknown central direction

“…f0;2 test when f 0 is posited to be f VMκ (von Mises with any valid value of κ), f C0.45 or f WC0.5 and the φ * (n);μ (n) 2 test (φ * (n);µ 2 with µ estimated from the data), calculated using 1000 samples of size n simulated from the k -sine-skewed distribution with the specified base density g 0 and values of λ and k . Table D.18: Rejection rates, for a nominal significance level of α = 0.05, of the b * 2 based andb 2 based tests calculated using 1000 samples of size n simulated from: the Kato and Jones (2010) distribution with parameters µ = 0, r = 0.5 and the values of ν and κ specified (KJ 10 ); the three-parameter asymmetric submodel given in the Equation (7) of Kato and Jones (2015) with parameters µ = 0, r = 0.5 and the values of γ andβ 2 = νγ(1 − γ) specified (KJ 15 ).…”

A random‐effects model for clustered circular data

“…The von Mises maximum likelihood estimator of β is therefore robust to a mis‐specification of the error distribution. This discussion does not apply to random effects with an asymmetric distribution, such as the family of Kato & Jones () or the sine skewed von Mises distribution of Abe & Pewsey (). The sampling properties of in models with asymmetric random effects are studied in Section 5, using Monte Carlo methods.…”

“…Additional simulations with a i and e ij having either a Kato–Jones distribution (Kato & Jones, ) or a von Mises distribution, were carried out to investigate the impact of mis‐specified random effects on the inference for β (see Table S2 of the Supplementary Material). The Kato–Jones family was selected because it can be asymmetric, with tails heavier than the von Mises as it generalizes the wrapped Cauchy distribution.…”

Smooth Kernel Estimation of a Circular Density Function: A Connection to Orthogonal Polynomials on the Unit Circle

“…More recently various alternatives to these classical parametric models, exhibiting asymmetry and multimodality, have been investigated with respect to their mathematical properties and goodness of fit to some real data; see Abe and Pewsey [5], Jones and Pewsey [6], Kato and Jones [7], Kato and Jones [8], Minh and Farnum [9], and Shimizu and Lida [10].…”