Linear Extension of the Yule Process
Yves Le Gat · 2015
The general formula established for the conditional probability of the NHBP is nevertheless not tractable for practical applications. We can somewhat sacrifice the generality to obtain a more practical result, by defining a Markovian process with intensity at t that linearly depends on the value reached by the counting process at t-. Such a linear dependency is shown to generate a counting process with a negative binomial distribution. This is called LEYP, which is the particular case of NHBP, for which the intensity linearly depends on N (t-). The chapter then considers the conditional distribution of the number of events likely to occur in a given time interval, which the authors call prediction interval or prediction window, given the number of events that happened in a previous time interval. This conditional distribution is envisaged in increasingly general configurations. The chapter also investigates the limiting behavior of the distributions when 945.