Research Priorities

• Quasi stationarity in Markovian models - can we model quasi stationarity and the risk of extinction in populations that can be modelling using reducible Markov chains?
• Statistical methods - can we determine robust and efficient estimation procedures for the location, scale and shape of probability distributions?

Researchers

• Chief Investigator: Phil Pollett
• Research Fellow: Iadine Chades
• Research Fellow: Ross McVinish
• Visiting Fellow: Olena Kravchuk
• PhD student : Daniel Pagendam
• PhD student : Thomas Taimre
• Honours student: Robert Cope

Collaborating Researchers

• Dr Jean Hu, Northwestern University
• Dr Olena Kravchuk, University of Queensland
• Prof Erik van Doorn, University of Twente

Research Projects Completed

• Quasi stationarity and survival risk in reducible Markovian models

  Project leader: Phil Pollett
  Researchers: Erik van Doorn (University of Twente)

  We have studied Markov chains in continuous time with a single absorbing state and a finite set S of transient states. When S is irreducible the limiting distribution of the chain, conditional on survival, is known to equal the (unique) quasi-stationary distribution of the chain. We addressed the problem of generalizing this result to a setting in which S may be reducible, and proved that it remains valid if the eigenvalue with maximal real part of the generator of the (sub) Markov chain on S has geometric (but not, necessarily, algebraic) multiplicity one. Our result was applied to pure death processes and, more generally, to quasi-death processes. Using classical theorems on M-matrices we showed that our result holds true even when the geometric multiplicity is larger than one, provided the irreducible subsets of S satisfy an accessibility constraint.

  Research outputs

  Van Doorn, E.A. and P.K. Pollett (2008) Survival in a quasi-death process. Linear Algebra and its Applications 429, 776-791.

• Estimating location, scale and shape of the generalized secant hyperbolic distribution

  Project leader: Phil Pollett
  Researchers: Olena Kravchuk (UQ), Jean Hu (Northwestern University)

  The generalized secant hyperbolic distribution (GSHD) was recently introduced as a modeling tool in data analysis. The GSHD is a unimodal distribution that is completely specified by location, scale and shape parameters. It has also been shown that the rank procedures of location are regular, robust, and asymptotically fully efficient. We studied certain tail weight measures for the GSHD and introduced a tail-adaptive rank procedure of location based on those tail weight measures. We investigated properties of the new adaptive rank procedure and compared it to some conventional estimators.

  Research outputs

  Kravchuk, O. and J. Hu (2008) Tail-adaptive location rank test of the generalized secant hyperbolic distribution. Communications in Statistics - Simulation and Computation 37, 1052-1063.

Awards and Achievements

• Honours student Robert Cope was awarded a travel scholarship from Emmanuel College, University of Queensland, to attend the 38th International Probability Summer School (Saint-Flour, France, 6-19 July 2008)
• AMSI-MASCOS PhD Scholar Daniel Pagendam was awarded a travel scholarship from the Isaac Newton Institute for Mathematical Sciences to attend the workshop "Designed Experiments: Recent Advances in Methods and Applications" (Cambridge, UK, 11-14 August 2008)
• Phil Pollett (with Hugh Possingham, The Ecology Centre, The University of Queensland) was awarded $71,346 from the Australian Centre of Excellence for Risk Analysis (ACERA) for a project titled "Strategies for managing invasive species in space: deciding whether to eradicate, contain or control" (2008-2009)
• AMSI-MASCOS PhD Scholar David Sirl completed his PhD degree (awarded May 2008) at the University of Queensland: thesis title "On the Analysis of Absorbing Markov Processes" (David currently holds a Research Fellowship (EPSRC) at the University of Nottingham)