Consistency of Markov chain quasi-Monte Carlo on continuous state spaces

The random numbers driving Markov chain Monte Carlo (MCMC) simulation are usually modeled as independent U(0,1) random variables. Tribble [Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences (2007) Stanford Univ.] reports substantial improvements when those random numbers are replaced by carefully balanced inputs from completely uniformly distributed sequences. The previous theoretical justification for using anything other than i.i.d. U(0,1) points shows consistency for estimated means, but only applies for discrete stationary distributions. We extend those results to some MCMC algorithms for continuous stationary distributions. The main motivation is the search for quasi-Monte Carlo versions of MCMC. As a side benefit, the results also establish consistency for the usual method of using pseudo-random numbers in place of random ones.Comment: Published in at http://dx.doi.org/10.1214/10-AOS831 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Longitudinal Joint Modelling of Binary and Continuous Outcomes: A Comparison of Bridge and Normal Distributions

Background: Longitudinal joint models consider the variation caused by repeated measurements over time as well as the association among the response variables. In the case of combining binary and continuous response variables using generalized linear mixed models, integrating over a normally distributed random intercept in the binary logistic regression sub-model does not yield to a closed form. In this paper, we assessed the impact of assuming a Bridge distribution for the random intercept in the binary logistic regression submodel and compared the results to that of normal distribution.  Method: The response variables are combined through correlated random intercepts. The random intercept in the continuous outcome submodel follows a normal distribution. The random intercept in the binary outcome submodel follows a normal or Bridge distribution. The estimations were carried out using a likelihood-based approach in direct and conditional joint modeling approaches. To illustrate the performance of the models, a simulation study was conducted Results: Based on the simulation results and regardless of the joint modeling approach, the models with a Bridge distribution for the random intercept of the binary outcome resulted in a slightly more accurate estimations and better performance. Conclusion: In addition to the fact that assuming a bridge distribution for the random intercept in binary logistic regression yields to the same interpretation of parameter estimates in marginal and conditional forms, our study revealed that even if the random intercept of binary logistic regression is normally distributed, assuming a Bridge distribution in the model will result in more accurate results.&nbsp

Analysis on flood generation processes by means of a continuous simulation model

International audienceIn the present research, we exploited a continuous hydrological simulation to investigate on key variables responsible of flood peak formation. With this purpose, a distributed hydrological model (DREAM) is used in cascade with a rainfall generator (IRP-Iterated Random Pulse) to simulate a large number of extreme events providing insight into the main controls of flood generation mechanisms. Investigated variables are those used in theoretically derived probability distribution of floods based on the concept of partial contributing area (e.g. Iacobellis and Fiorentino, 2000). The continuous simulation model is used to investigate on the hydrological losses occurring during extreme events, the variability of the source area contributing to the flood peak and its lag-time. Results suggest interesting simplification for the theoretical probability distribution of floods according to the different climatic and geomorfologic environments. The study is applied to two basins located in Southern Italy with different climatic characteristics

Cramer type moderate deviations for random fields and mutual information estimation for mixed-pair random variables

In this dissertation we first study Cramer type moderate deviation for partial sums of random fields by applying the conjugate method. In 1938 Cramer published his results on large deviations of sums of i.i.d. random variables after which a lot of research has been done on establishing Cramer type moderate and large deviation theorems for different types of random variables and for various statistics. In particular results have been obtained for independent non-identically distributed random variables for the sum of independent random to estimate the mutual information between two random variables. The estimates enjoy a central limit theorem under some regular conditions on the distributions. The theoretical results are demonstrated by simulation study. variables with p-th moment (p \u3e 2) and for different types of dependent random variables. In this work we establish Cramer type exact moderate deviation theorem for random fields. We then show that obtained results are applicable to the partial sums of linear random fields with short or long memory and to non-parametric regression with random field errors. We also show that the result for linear random fields can be applied to calculate the tail probability of partial sums of various models such as the autoregressive fractionally integrated moving average FARIMA(p;\beta; q) processes. The results can also be used to approximate the risk measures such as quantiles and tail conditional expectations of time series or spacial random fields. We also study the mutual information estimation for mixed-pair random variables. One random variable is discrete and the other one is continuous. We develop a kernel metho