Measuring Skewness for Circular Distributions

• Activity Number: 663
• Type: Contributed
• Date/Time: Thursday, August 4, 2011 : 10:30 AM to 12:20 PM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Dale Umbach*+ and S. Rao Jammalamadaka
• Companies: Ball State University and University of California at Santa Barbara
• Keywords: circular distribution ; skewness ; trigionometric moments ; skew-symmetric

Abstract

We consider the problem of measuring the skewness of circular distributions through two approaches. The first is based on the fact that the pth central trigonometric moment of a symmetric circular distribution is real, i.e. for a circular random variable T with circular mean µ, the imaginary components ß*p=E[sin(p(T- µ))] = 0 for p=0, ±1, ±2, (See e.g. Jammalamadaka and SenGupta (2001, Topics in Circular Statistics). This suggests that one could use the imaginary components of these moments in measuring skewness of a distribution. Indeed, Batschelet (1965, American Institute of Biological Sciences) introduced ß*2 as a measure of skew, which Pewsey (2002, Canadian Journal of Statistics) used for testing for skewness. We will consider other trigonometric moment based measures besides ß*2. Many percentile based measures of skewness have been studied for the linear case, see e.g. Groeneveld and Meeden (2009, Metron). Such measures are dependent on the choice of origin. Instead, we propose t? = (Pr(µ < T = µ + ?) - Pr(µ - ? = T < µ))/ Pr(µ - ? = T = µ + ?) as a measure of skewness for various values of 0< ? < p. This measure is independent of the choice of origin.