Publikationsserver der RWTH Aachen University
Markovian simple counting processes and models of ordered Random variables
Abstract
dc:descriptionA new definition of pure birth processes, including the usual definition, but based on qualities of the underlying increasing point process, is proposed. In the new definition (in contrast to the usual definition) the transition probabilities of the counting process are allowed to fail to be absolutely continuous, for the proposed pure birth-Bernoulli process they are even allowed to be discontinuous (corresponding to fixed jumps in the counting process). The idea underlying the (new) construction of pure birth processes is a characterization of inhomogeneous Poisson processes with continuous mean value function as a time-transformed standard Poisson process, where in the (new) definition of pure birth processes that time-transformation is allowed to depend on the state of the process. This new definition (together with its new point of view) allows elegant and simple proofs of known results, which often can be generalized to less rigid assumptions, and lead to deeper insights than the prominent calculations. Especially we are interested in the Markovian structure of pure birth processes and models of ordered random variables. We prove that the jump times of pure birth processes coincide (in distribution) with Pfeifers record values (record values where the distribution of the underlying random variables may change after each record) and sequential order statistics (a model to analyse the reliability of certain k-out-of-n systems where the distribution of failures may depend on the number of failed elements), each for continuous distribution functions for the underlying random variables. Pure birth processes with operational time coincide (in distribution) with generalized order statistics (also a model for the reliability of certain k-out-of-n systems with more restrictions on the dependency of the number of failed elements) for continuous distribution functions for the underlying random variables. This model also includes other models of random variables (by a certain choice of parameters) like record values, order statistics, and progressively Type II censored order statistics. Furthermore and with similar methods as in the continuous case the pure birth-Bernoulli process is introduced as a pure birth process with fixed jumps. The coincidence in distribution remains valid for pure birth-Bernoulli processes and Pfeifer records (with possibly discontinuous distribution functions for the underlying random variables), and for pure birth-Bernoulli processes and sequential order statistics with weakly increasing support (and possibly discontinuous distribution functions for the underlying random variables). Also, the jump times of a Poisson-Bernoulli process (as special case of the pure birth-Bernoulli process) coincide (in distribution) with record values from random variables with possibly discontinuous distribution function. Finally we analyse and compare distinguished subclasses: Mixed Poisson processes as a subclass of pure birth processes, where the corresponding counterpart in models of ordered random variables for a Gamma-mixing variable are m-generalized order statistics with m<-1, where m=-1 corresponds to a degenerate mixing variable. Interesting subclasses for generalized order statistics are those with an increasing chain of parameters and decreasing chain of parameters, where the first possibility leads to an over-dispersed pure birth process with positive contagion, the second to an under-dispersed pure birth process with negative contagion.
Degree
thesis:*- Grantor dc:publisher
- Publikationsserver der RWTH Aachen University
- Year dc:date
- 2008
Author and committee
dc:creator, dc:contributor.*- Author dc:creator
-
- Lenz, Björn
- Contributors dc:contributor
-
- Kamps, Udo
Subjects
dc:subject × 8Rights
dc:rights- Statement dc:rights
-
- info:eu-repo/semantics/openAccess
- Language dc:language
- eng