Walter Dempsey — Page 3

arXiv2015

The pilgrim process

Pilgrim's monopoly is a probabilistic process giving rise to a non-negative sequence $T_1, T_2,\ldots$ that is infinitely exchangeable, a natural model for time-to-event data. The one-dimensional marginal distributions are exponential. The rules are simple, the process is easy to generate sequentially, and a simple expression is available for both the joint density and the multivariate survivor function. There is a close connection with the Kaplan-Meier estimator of the survival distribution. Embedded within the process is an infinitely exchangeable ordered partition processes connected to Markov branching processes in neutral evolutionary theory. Some aspects of the process, such as the distribution of the number of blocks, can be investigated analytically and confirmed by simulation. By ignoring the order, the embedded process can be considered as an infinitely exchangeable partition process, shown to be closely related to the Chinese restaurant process. Further connection to the Indian buffet process is also provided. Thus we establish a previously unknown link between the well-known Kaplan-Meier estimator and the important Ewens sampling formula.

math.ST↗

arXiv2015

Weak continuity of predictive distribution for Markov survival processes

We explore the concept of a consistent exchangeable survival process - a joint distribution of survival times in which the risk set evolves as a continuous-time Markov process with homogeneous transition rates. We show a correspondence with the de Finetti approach of constructing an exchangeable survival process by generating iid survival times conditional on a completely independent hazard measure. We describe several specific processes, showing how the number of blocks of tied failure times grows asymptotically with the number of individuals in each case. In particular, we show that the set of Markov survival processes with weakly continuous predictive distributions can be characterized by a two-dimensional family called the harmonic process. We end by applying these methods to data, showing how they can be easily extended to handle censoring.

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