Volume 15, Issue 2

The famous Propp and Wilson (Random Structures and Algorithms 9: 223–252, 1996, Journal of the American Statistical Association 90: 558–566, 1998) protocol called coupling from the past (CFTP) allows exact sampling from steady-state distribution of a Markov chain. When the Markov chain is stiff (i.e. existence of rarely visited states), CFTP spends a prohibitive time to reach stationarity. To reduce this time we propose to combine the variance reduction technique Importance Sampling (IS) with CFTP. Also we propose another technique, based on the power of the Markov chain kernel, to reduce the CFTP simulation time in standard case. When the period δ of the simulated Markov chain is greater than one ( δ > 1), the stopping condition of CFTP is not satisfied. To break the deadlock of CFTP in this case, we propose to transform the studied chain on δ subchains that are aperiodic and for which CFTP can be applied. Some numerical examples are presented to bring the utility of the proposed simulation techniques.

The normalized exponential-normal form A · exp(–( x – c ) 2 /( a ( x – c ) + 2 b 2 )) is an asymmetric distribution intermediate between the normal and exponential distributions. Some properties of the form are presented and some methods of approximation are suggested. Appropriate formulae and table are presented. The quality of original data defines the specific features of the methods. Application of the methods is illustrated by ultra high energy atmospheric showers investigation. The relation to the exponential and normal distributions makes the form useful and effective in applications.

Importance sampling is a standard variance reduction tool in Monte Carlo integral evaluation. It postulates estimating the integrand just in the areas where it takes big values. It turns out this idea can be also applied to multivariate optimization problems if the objective function is non-negative. We can normalize it to a density function, and if we are able to simulate the resulting p.d.f., we can assess the maximum of the objective function from the respective sample.

Compartmental analysis is used to model dynamic biological systems and widely applied to study the kinetics of drugs in the body. We used compartmental mixed-effects modeling, which quantifies the between- and within-subject variability, to analyze population data based on a pharmacokinetic model where predictions were obtained from a solution of a system of ordinary differential equations (ODEs). Non-Bayesian software (nlmeODE in R, or, NLINMIX with ODEs in SAS) and Bayesian software (WBDiff in WinBUGS) enabled the mixed-effect analysis of complicated systems of ODEs with and without a closed-form solution. Our aim was to use simulation data and real data from the glucose-insulin minimal model study to illustrate the applicability of Bayesian and non-Bayesian methods for compartmental analysis of population data. Our results indicated that the two methods are numerically stable and provided accurate parameter estimates for the simulation data based on the standard PK/PD model. However, in the analysis of glucose-insulin minimal model, the Bayesian method was preferred, since it provided a satisfactory solution (statistically) to the minimal model without approximation by linearization required by the non-Bayesian algorithm.

We discuss an error in a proof on the discrepancy of mixed sequences (in [Ökten, Contributions to the theory of Monte Carlo and quasi-Monte Carlo methods, Ph.D. dissertation, The Claremont Graduate School, 1997, Ökten, Monte Carlo Methods and Applications 2: 255–270, 1996]) and recent research ([Gnewuch, Journal of Complexity 25: 312-317, 2009, Ökten, Tuffin, Burago, Journal of Complexity 22: 435–458, 2006]) that provide corrections, as well as research that provide a different generalization of the mixed sequences.