The Semicircular Reflected Gamma Distribution

Quite a lot of work was done on circular models defined on the unit circle (Jammalamadaka and Sengupta 2001); (Mardia and Jupp, 2000) . In some cases, the directional/angular data do not require full circular models for modeling, this fact is noted in Jones (1968), Guardiola (2004 ), Byoung (2008 ) and Phani et al. (2013a). For example, when sea turtles emerge from the ocean in search of a nesting site on dry land, a random variable having values on a semicircle is well sufficient for modeling such data. Similarly, when an aircraft is lost but its departure and its initial headings are known, a semicircular random variable is sufficient for such angular data. And few more examples of semicircular data are available in Ugai et.al, (1977). Guardiola ( 2004 ) has obtained the semicircular normal distribution by using a simple projection and Byoung (2008) have developed a family of semicircular Laplace distributions for modeling semicircular data by simple projection. Phani et al. (2013a) constructed some semicircular distributions by applying modified inverse stereographic projection. In this paper, the authors derive the Semicircular Reflected Gamma distribution by projecting Reflected Gamma distribution over a semicircular segment; this distribution generalizes semicircular Laplace distribution (Byoung , 2008). The authors have plot the graphs of the density function for various values of the parameters. Furthermore, the first four trigonometric moments for proposed model are derived and also they extend this model for l-axial Reflected Gamma distribution for modelling any arc of arbitrary length. Projection is defined by a one-to-one mapping given by, THE SEMICIRCULAR REFLECTED GAMMA DISTRIBUTION By PHANI YEDLAPALLI