No. 1088

Title:

On the conditional and unconditional distributions of the number of success runs on a circle with applications

Author(s):

Kiyoshi, Inoue (Faculty of Economics, Seikei University);
Sigeo, Aki (Faculty of Engineering, Kansai University)

Key words:

Binary trials, circular success runs, enumeration schemes, exchangeable sequence, circular binomial distribution of order k, probability function, moments, conditional probability function, run test, reliability, Pólya urn model

Abstract:

    In this paper, we study distributions of the number of success runs of length k and the number of success runs of length k given the number of successes in a sequence of independent and identically distributed (i.i.d.) binary trials arranged on a circle (circular sequence) based on three different enumeration schemes. The double generating functions, the probability functions and a formula for the evaluation of the higher order moments are given. Furthermore, we show that the results established in the case of i.i.d. circular sequence could be extended for the study of the distribution of number of success runs in a circular sequence of binary exchangeable trials. We offer tools for the run-related problems arising from the circular exchangeable sequence. Some applications to practical problems such as statistical run tests for exchangeability, reliability theory and Pólya urn models are given in order to show our theoretical results, which illustrate the potential use of run statistics.