Location of Repository

Many Markov chains with a single absorbing state have a unique limiting conditional distribution (LCD) to which they converge, conditioned on non-absorption, regardless of the initial distribution. If this limiting conditional distribution is used as the initial distribution over the non-absorbing states, then the probability distribution of the process at time n, conditioned on non-absorption, is equal for all values of n>0. Such an initial distribution is known as the quasi-stationary distribution (QSD). Thus the LCD and QSD are equal. These distributions can be found in both the discrete-time and continuous-time case.\ud \ud In this thesis we consider finite Markov chains which have one absorbing state, and for which all other states form a set which is a single communicating class. In addition, every state is aperiodic. These conditions ensure the existence of a unique LCD. We first consider continuous Markov chains in the context of survival analysis. We consider the hazard rate, a function which measures the risk of instantaneous failure of a system at time t conditioned on the system not having failed before t. It is well-known that the QSD leads to a constant hazard rate, and that the hazard rate generated by any other initial distribution tends to that constant rate. Claims have been made by Aalen and by Aalen and Gjessing that it may be possible to predict the shape of hazard rates generated by phase type distributions (first passage time distributions generated by atomic initial distributions) by comparing these initial distributions with the QSD. In Chapter 2 we consider these claims, and demonstrate through the use of several examples that the behaviour considered by those conjectures is more complex then previously believed.\ud \ud In Chapters 3 and 4 we consider discrete Markov chains in the context of imprecise probability. In many situations it may be unrealistic to assume that the transition matrix of a Markov chain can be determined exactly. It may be more plausible to determine upper and lower bounds upon each element, or even determine closed sets of probability distributions to which the rows of the matrix may belong. Such methods have been discussed by Kozine and Utkin and by Skulj, and in each of these papers results were given regarding the long-term behaviour of such processes. None of these papers considered Markov chains with an absorbing state. In Chapter 3 we demonstrate that, under the assumption that the transition matrix cannot change from time step to time step, there exists an imprecise generalisation to both the LCD and the QSD, and that these two generalisations are equal. In Chapter 4, we prove that this result holds even when we no longer assume that the transition matrix cannot change from time step to time step. In each chapter, examples are presented demonstrating the convergence of such processes, and Chapter 4 includes a comparison between the two methods

Topics:
Hazard rate, imprecise probability, Markov chains

Year: 2009

OAI identifier:
oai:etheses.dur.ac.uk:14

Provided by:
Durham e-Theses

Downloaded from
http://etheses.dur.ac.uk/14/1/thesis.pdf

- (1995). A ratio limit theorem for (sub)
- (1995). A ratio limit theorem for (sub) Markov chains on 1,2,... with bounded jumps.
- (1981). A Second Course in Stochastic Processes.
- (1978). An Introduction to Orthogonal Polynomials.
- (1950). An Introduction to Probability Theory and its Applications, Volume 1, 3rd Edition.
- (1980). Bathtub and related failure rate characterizations,
- (1994). Bayesian reliability analysis with imprecise prior probabilities.
- (2003). Beta-invariant measures for transition matrices of G1/M/1 type.
- (1947). Certain limit theorems of the theory of branching stochastic processes (in Russian).
- (1981). Coherent lower (and upper) probabilities.
- (1991). Continuous-Time Markov Chains: an ApplicationsOriented Approach.
- (1970). Counterexamples in Topology.
- (2001). Elementare Grundbegriﬀe einer allgemeineren Wahrscheinlichkeitsrechnung I. Intervallwahrscheinlichkeit als umfassendes Konzept.
- (1996). Evolution with state-dependent mutations.
- (2006). Finite discrete time Markov chains with interval probabilities.
- (2002). Fuzzy Markov chains.
- (1995). Geometric ergodicity and quasistationarity in discrete-time birth-death processes.
- (1986). Hazard-rate analysis in stage I malignant melanoma.
- (2009). Imprecise Markov chains and their limit behaviour. Pre-print at arXiv.org,
- (2009). Imprecise Markov chains with absorption. In submission.
- (2009). Imprecise Markov chains with an absorbing state, In
- (2002). Interval-valued ﬁnite Markov chains.
- (1987). Limit theorems for the population size of a birth and death process allowing catastrophes.
- (1972). Log-concavity and log-convexity in passage time densities of diﬀusion and birth-death processes.
- (1979). Markov chain models - rarity and exponentiality.
- (1973). Markov chains and M-matrices: inequalities and equalities.
- (1997). Markov Processes for Stochastic Modeling.
- (1998). Markov Set Chains.
- (1981). Note on the theory of a self-catalytic chemical reaction,
- (1995). od Discrete-Time Markov Chains.
- (1966). On quasi-stationary distributions in discrete-time Markov chains with a denumerable inﬁnity of states.
- (1967). On quasistationary distributions in absorbing continuous-time ﬁnite Markov chains.
- (1965). On quasistationary distributions in absorbing discrete-time ﬁnite Markov chains.
- (1960). On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov process. ˇ Casopis pro pestovani matematiky a fyziky 85,
- (1960). On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov process. Cˇasopis pro pestovani matematiky a fyziky 85,
- (1956). On the composition of unimodal distributions. Theory of Probability and its Applications,
- (1981). On the unimodality of passage time densities in birth-death processes.
- (1975). Optimal rules for ordering uncertain prospects.
- (1968). Perturbation theory and ﬁnite Markov chains.
- (1995). Phase type distributions in survival analysis.
- (1995). Probability and Measures.
- (1992). Probability and Random Processes.
- (1975). Probability distributions of phase type. Liber Amicorum Prof. Emeritus H.
- (2005). Probability, Statistics, and Stochasitc Processes.
- (1995). Quasi-stationarity od Discrete-Time Markov Chains.
- (1966). Quasi-stationary behaviour in the random walk with continuous time.
- (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes.
- (2006). Quasi-stationary distributions for a class of discrete-time
- (1987). Quasistationary distributions for autocatalytic reactions.
- (1995). Ratio limits and limiting conditional distributions for discrete-time birth-death processes.
- (2007). Regular ﬁnite Markov chains with interval probabilities.
- (1962). Renewal Theory.
- (2007). Semi-parametric Bayesian analysis of the proportional hazard rate model: an application to the eﬀect of training programs on graduate unemployment.
- (1989). Spectral theory for skip-free Markov chains.
- (1954). Spectral theory for the diﬀerential equations of simple birth-and-death processes.
- (1991). Statistical Reasoning with Imprecise Probabilities.
- (2008). Stochastic stability and time-dependent mutations.
- (1996). Survival Analysis With Long-Term Survivors.
- (2008). Survival and Event History Analysis: A Process Point of View,
- (1957). The classiﬁcation of birth and death processes.
- (1957). The diﬀerential equations of birth-death processes, and the Stieltjes Moment Problem.
- (1998). The Inverse Gaussian Distribution: Statistical Theory and Applications.
- (1989). The Inverse Gaussian Distribution: Theory, Methodology and Applications.
- (1962). The relationship between the mean and variance of a stationary birth-death process, and its economic application.
- (1951). The rise and fall of a reindeer herd,
- (2000). The theory of interval-probability as a unifying concept for uncertainty.
- (2009). Time homogeneous birth-death processes with probability intervals and absorbing state.
- (1993). Topologies On Closed And Closed Convex Sets.
- (2001). Understanding the shape of the hazard rate: a process point of view.
- (1997). Weak convergence of conditioned birth-death processes in discrete time.
- (2003). Weakly Symmetric Graphs, Elementary Landscapes and the TSP.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.