The Mobius Distribution on the Disc

i . Introduction How many probabi l i ty dis t r ibut ions can you th ink of wi th suppor t the unit disc? Th e uniform dis t r ibut ion probably comes to mind immediately. Th e only other dis t r ibut ions with suppor t the uni t disc tha t have any degree of familiari ty are the bivariate spherically symmetr ic be ta (or Pearson type II) dis t r ibut ions wi th densi ty (1.1) fir(x, y) = ~ (1 x 2 y2) r -1 ")' > 0, 0 < x 2 + y2 ~ 1. These distr ibutions include the uniform dis t r ibut ion on the disc (sometimes called the 'circular law') when V = 1 and are all spherically symmetr ic with mode at zero when V > 1 and ant imode at zero when V < 1. Marginals are symmetr ic be ta dis tr ibut ions on [ -1 , 1] with pa rame te r V + 1/2. See Kotz (1975), Johnson (1987) and Fang et al. (1990) for details. But these dis tr ibut ions are all spherically symmetr ic . How about dis tr ibut ions tha t can also exhibit skewness, i.e., t ha t do not exhibit bivariate s y m m e t r y on the disc? Th e only specific proposal tha t I know of was made by Jones ((2002), Section 3). Bu t now I can do bet ter! In this paper , I in t roduce an a l ternat ive simple family of dis t r ibut ions on the disc which has many skew members and which also incorporates the be ta d is t r ibut ion (1.1) as its spherically symmetr ic special case. In addit ion, all densi ty functions in the new family remain unimodal , mono tone or uniant imodal . Other advantages include meaningful pa ramete rs tha t are readi ly es t imated by m ax im u m likelihood and which prove to exhibit pa ramete r or thogonal i ty .