Volume 30, Issue 2

A Hidden Markov Model (HMM) is a couple of stochastic processes: A hidden Markov process and an observed emission process. Generally, the HMMs are used to study the hidden behavior of random systems through some observed emission sequences generated by the phenomenon under study. In this frame-work, we propose to solve the likelihood and the decoding problems of HMMs whose state space is composed of a continuous part and a discrete part. We adapt forward, backward and Viterbi algorithms to the case of our proposal. Numerical examples and Monte Carlo simulations are considered to show the efficiency and the adaptation of the algorithms for the proposed model.

It is well known that shortened modeling of particle trajectories with the use multiplicative statistical weights, as a rule, increases the efficiency of the program (in terms of accuracy/time ratio). This trick is often used in non-branching schemes simulating transfer processes without multiplication (for example, the transfer of X-ray radiation), in which it is sufficient to confine ourselves to studying only the average values of the field characteristics. With an increase in energy, however, multiplication processes begin to play a significant role (the production of electron-photon pairs by gamma quanta with energies above 1.022 MeV, etc.), when the resulting trajectory is not just a broken curve in the phase space, but a branched tree. This technique is also applicable to this process, but only if the study of statistical fluctuations and correlations is not the purpose of the calculation. The present review contains the basic concepts of the Monte Carlo method as applied to the theory of particle transport, demonstration of the weighting method in non-branching processes, and ends with a discussion of unbiased estimates of the second moment and covariance of additive functionals.

This paper considers three approaches to choosing the constant H in the expression H ⁢ 𝐃 ⁢ ζ / n {H\sqrt{{\mathbf{D}}\zeta}/\sqrt{n}} for the error of the Monte Carlo method for numerical calculation of mathematical expectation 𝐄 ⁢ ζ {{\mathbf{E}}\zeta} of a random variable ζ: in probability, in mean square and in mean.In practical studies using the Monte Carlo method, when estimating the calculation error, it is recommended to use the “in mean” approach with the constant H = 2 π = 0.79788456079 ⁢ … {H=\sqrt{\frac{2}{\pi}}=0.79788456079\dots}   .

In this paper, we study linear regression models in which the error term has shape mixtures of skew-normal distribution. This type of distribution belongs to the skew-normal (SN) distribution class that can be used for heavy tails and asymmetry data. For the first time, for the classical (non-Bayesian) estimation of the parameters of the SN family, we apply the Markov chains Monte Carlo ECM (MCMC-ECM) algorithm where the samples are generated by Gibbs sampling, denoted by Gibbs-ECM, and also, we extend two other types of the EM algorithm for the above model. Finally, the proposed method is evaluated through a simulation and compared with the Numerical Math-ECM algorithm and Monte Carlo ECM (MC-ECM) using a real data set.

In this paper, we consider the filtering of partially observed multi-dimensional diffusion processes that are observed regularly at discrete times. We assume that, for numerical reasons, one has to time-discretize the diffusion process, which typically leads to filtering that is subject to discretization bias. The approach in [A. Jasra, K. J. H. Law and F. Yu, Unbiased filtering of a class of partially observed diffusions, Adv. Appl. Probab. 54 (2022), 3, 661–687] establishes that, when only having access to the time discretized diffusion, it is possible to remove the discretization bias with an estimator of finite variance. We improve on this method by introducing a modified estimator based on the recent work [A. Jasra, M. Maama and H. Ombao, Antithetic multilevel particle filters, preprint (2023), https://arxiv.org/abs/2301.12371]. We show that this new estimator is unbiased and has finite variance. Moreover, we conjecture and verify in numerical simulations that substantial gains are obtained. That is, for a given mean square error (MSE) and a particular class of multi-dimensional diffusion, the cost to achieve the said MSE falls.

Classical jittered sampling partitions [ 0 , 1 ] d {[0,1]^{d}} into m d {m^{d}} cubes for a positive integer m and randomly places a point inside each of them, providing a point set of size N = m d {N=m^{d}} with small discrepancy. The aim of this note is to provide a construction of partitions that works for arbitrary N and improves straight-forward constructions. We show how to construct equivolume partitions of the d -dimensional unit cube with hyperplanes that are orthogonal to the main diagonal of the cube. We investigate the discrepancy of such point sets and optimise the expected discrepancy numerically by relaxing the equivolume constraint using different black-box optimisation techniques.

We develop a Monte Carlo method to solve backward stochastic differential equations (BSDEs) in high dimensions. The proposed algorithm is based on the regression-later approach using multivariate Hermite polynomials and their gradients. We propose numerical experiments to illustrate its performance.

Multi-stage models for cohort data are popular statistical models in several fields such as disease progressions, biological development of plants and animals, and laboratory studies of life cycle development. A Bayesian approach on adopting deterministic transformations in the Metropolis–Hastings (MH) algorithm was used to estimate parameters for these stage structured models. However, the biases in later stages are limitations of this methodology, especially the accuracy of estimates for the models having more than three stages. The main aim of this paper is to reduce these biases in parameter estimation. In particular, we conjoin insignificant previous stages or negligible later stages to estimate parameters of a desired stage, while an adjusted MH algorithm based on deterministic transformations is applied for the non-hazard rate models. This means that current stage parameters are estimated separately from the information of its later stages. This proposed method is validated in simulation studies and applied for a case study of the incubation period of COVID-19. The results show that the proposed methods could reduce the biases in later stages for estimates in stage structured models, and the results of the case study can be regarded as a valuable continuation of pandemic prevention.