Volume 18, Issue 2

Abstract. In this paper, the robust extreme ranked set sampling (RERSS) scheme is considered for estimating the population median. The RERSS is compared with the simple random sampling (SRS), ranked set sampling (RSS) and extreme ranked set sampling (ERSS) schemes. A Monte Carlo simulation study is used to study the performance of the median estimator. It is found that RERSS estimators are unbiased of the population median when the underlying distribution is symmetric. Also, in terms of the efficiency criterion; the median estimator based on RERSS is more efficient than the median estimators based on SRS, ERSS, and RSS for symmetric and asymmetric distributions considered in this study. For asymmetric distributions, the RERSS estimators have a smaller bias.

Abstract. We propose a mathematical analysis of a well-known numerical approach used in molecular dynamics to efficiently sample a coarse-grained description of the original trajectory (in terms of state-to-state dynamics). This technique is called parallel replica dynamics and has been introduced by Arthur F. Voter. The principle is to introduce many replicas of the original dynamics, and to consider the first transition event observed among all the replicas. The effective physical time is obtained by summing up all the times elapsed for all replicas. Using a parallel implementation, a speed-up of the order of the number of replicas can thus be obtained, allowing longer time scales to be computed. By drawing connections with the theory of Markov processes and, in particular, exploiting the notion of quasi-stationary distribution, we provide a mathematical setting appropriate for assessing theoretically the performance of the approach, and possibly improving it.

Abstract. In this paper, we describe a stochastic algorithm to solve sudoku puzzles. Our method consists in computing probabilities for each symbol of each cell updated at each step of the algorithm using estimation of distributions algorithms (EDA). This update is done using the empirical estimators of these probabilities for a fraction of the best puzzles according to a cost function. We develop also some partial restart techniques in the RESEDA algorithm to obtain a convergence for the most difficult puzzles. Our algorithm is tested numerically on puzzles with various levels of difficulty starting from very easy ones to very hard ones including the famous puzzle AI Escargot. The CPU times vary from few hundreds of a second for the easy ones to about one minute for the most difficult one.

Abstract. Parallel pseudorandom number generators (PPRNG) that satisfy classical statistical tests may still demonstrate intra-stream and inter-stream correlations in real life applications. In order to investigate the suitability of a PPRNG for use in Monte Carlo pricing of financial derivatives, an application-based test is proposed to evaluate the bias and the standard error of the mean (SE) associated with the PPRNG as a gauge of intra-stream and inter-stream correlations respectively. This test involves estimating the price of a vanilla European call option via Monte Carlo simulation, where the asset price at maturity is estimated by propagating the Black–Scholes stochastic differential equation via the Euler–Maruyama discretization scheme. The mean and SE profiles of the numerical results based on three PPRNG libraries ( RngStream , TRNG and SPRNG ) that implement parallel random numbers via sequence splitting strategies ( RngStream and TRNG ) and parameterization strategy ( SPRNG ) are compared. In terms of the bias and SE profiles, the best performing PPRNG constructed using the sequence splitting strategy is comparable to that constructed using parameterization, both use multiple recursive generators in their kernel.

Abstract. In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and QMC, are so-called mixed sequences, which are constructed by concatenating a d -dimensional QMC sequence and an ()-dimensional MC sequence to obtain a sequence in dimension s . Ökten, Tuffin and Burago proved probabilistic asymptotic bounds for the discrepancy of mixed sequences, which were refined by Gnewuch. In this paper we use an interval partitioning technique to obtain improved probabilistic bounds for the discrepancy of mixed sequences. By comparing them with lower bounds we show that our results are almost optimal.