Circular Polya distributions of order \(k\) (Q1811930)

Summary: Two circular Polya distributions of order \(k\) are derived by means of generalized urn models and by compounding, respectively, the type I and type II circular binomial distributions of order \(k\) of \textit{F.S. Makri} and \textit{A.N. Philippou} [A.P. Godbole et al. (eds.), Runs and Patterns in Probability. Math. Appl., Dordr. 283, 65-81 (1994; Zbl 0831.60094)] with the beta distribution. It is noted that the above two distributions include, as special cases, new circular hypergeometric, negative hypergeometric, and discrete uniform distributions of the same order and type. The means of the new distributions are obtained and two asymptotic results are established relating them to the above-mentioned circular binomial distributions of order \(k\).