Inverse stereographic hyperbolic secant distribution: a new symmetric circular model by rotated bilinear transformations

Abdullah Yilmaz

doi:10.31801/cfsuasmas.906339

Abstract

The inverse stereographic projection (ISP), or equivalently, bilinear transformation, is a method to produce a circular distribution based on an existing linear model. By the genesis of the ISP method, many important circular models have been provided by many researchers. In this study, we propose a new symmetric unimodal/bimodal circular distribution by the rotated ISP method considering the hyperbolic secant distribution as a baseline distribution. Rotation means that fixing the origin and rotating all other points the same amount counterclockwise. Considering the effect of rotation on the circular distribution to be obtained with the bilinear transformation, it is seen that it actually induces a location parameter in the obtained circular probability distribution. We analyze some of the stochastic properties of the proposed distribution. The methods for the parameter estimation of the new circular model and the simulation-based compare results of these estimators are extensively provided by the paper. Furthermore, we compare the fitting performance of the new model according to its well-known symmetric alternatives, such as Von-Misses, and wrapped Cauchy distributions, on a real data set. From the information obtained by the analysis on the real data, we say that the fitting performance of the new distribution is better than its alternatives according to the criteria frequently used in the literature.

Highlights

Circular or directional data are observed in various fields of science
The inverse stereographic projection (ISP), or equivalently, bilinear transformation, is a method to produce a circular distribution based on an existing linear model
The location parameter to be added to the circular probability distributions obtained by the ISP method corresponds to the rotation property of bilinear transforms

Summary

INTRODUCTION

Circular or directional data are observed in various fields of science. Data on angular observations can often be associated with compass measurements. By considering the Eq(8) and Eq(9) with Eq(12) and Eq(11), we obtain the cdf and pdf of the inverse stereographic hyperbolic secant distributed random variable. It is seen that the Bias and MSE values decrease to zero as the sample size increases for the estimation of parameters α and v by all three methods. This shows that the estimates are precise and accurate, we say that it is consistent and unbiased. In order to make comparisons, we chosed the Von-Mises (VM) and Wrapped Cauchy (WC) distributions as well-known alternatives from the location family for modeling symmetrical circular data.

Model Method

Conclusion