D. R. Cox, “Some Statistical Methods Connected with Series of Events,” Journal of the Royal Statistical Society B, Vol. 17, No. 2, 1955, pp. 129-164.

has been cited by the following article:

• TITLE: The Distribution of Multiple Shot Noise Process and Its Integral

  AUTHORS: Jiwook Jang

  KEYWORDS: Multiple Shot Noise Process and Its Integral; Multivariate Cox Process; Piecewise Deterministic Markov Process; Martingale Methodology; Multivariate Survival Functions

  JOURNAL NAME: Applied Mathematics, Vol.5 No.3, February 12, 2014

  ABSTRACT: In this paper, we study multiple shot noise process and its integral. We analyse these two processes systematically for their theoretical distributions, based on the piecewise deterministic Markov process theory developed by Davis [1] and the martingale methodology used by Dassios and Jang [2]. The analytic expressions of the Laplace transforms of these two processes are presented. We also obtain the multivariate probability generating function for the number of jumps, for which we use a multivariate Cox process. To derive these, we assume that the Cox processes jumps, intensity jumps and primary event jumps are independent of each other. Using the Laplace transform of the integral of multiple shot noise process, we obtain the tail of multivariate distributions of the first jump times of the Cox processes, i.e. the multivariate survival functions. Their numerical calculations and other relevant joint distributions’ numerical values are also presented.