Decay rates for quasi-birth-and-death processes with countably many phases and tridiagonal block generators

We consider the class of level-independent quasi-birth-and-death (QBD) processes that have countably many phases and generator matrices with tridiagonal blocks that are themselves tridiagonal and phase independent. We derive simple conditions for possible decay rates of the stationary distribution of the ‘level’ process. It may be possible to obtain decay rates satisfying these conditions by varying only the transition structure at level 0. Our results generalize those of Kroese, Scheinhardt, and Taylor, who studied in detail a particular example, the tandem Jackson network, from the class of QBD processes studied here. The conditions derived here are applied to three practical examples.

LASSO model selection with post-processing for a genome-wide association study data set

Model selection procedures for simultaneous analysis of all single-nucleotide polymorphisms in genome-wide association studies are most suitable for making full use of the data for a complex disease study. In this paper we consider a penalized regression using the LASSO procedure and show that post-processing of the penalized-regression results with subsequent stepwise selection may lead to improved identification of causal single-nucleotide polymorphisms.

Decay rates for some quasi-birth-and-death processes with phase-dependent transition rates

Recently, there has been considerable interest in the calculation of decay rates for models that can be viewed as quasi-birth-and-death (QBD) processes with infinitely many phases. In this paper we make a contribution to this endeavour by considering some classes of models in which the transition function is not homogeneous in the phase direction. We characterize the range of decay rates that are compatible with the dynamics of the process away from the boundary. In many cases, these rates can be attained by changing the transition structure of the QBD process at level 0. Our approach, which relies on the use of orthogonal polynomials, is an extension of that in Motyer and Taylor (2006) for the case where the generator has homogeneous blocks.