Abstract We show that Lasso and Bayesian Lasso are very close when the sparsity is large and the noise is small. We propose to solve Bayesian Lasso using multivalued stochastic differential equation. We derive four discretization algorithms, and present highly efficient multilevel Monte Carlo (MLMC) simulations. Additionally, we perform a numerical comparison of the Monte Carlo (MC), MLMC and proximal Markov chain Monte Carlo algorithm (PMALA).

Abstract This work suggests different Monte Carlo algorithms for solving large systems of linear algebraic equations arising from the numerical solution of the Dirichlet problem for the Helmholtz equation. Approach based on boundary integral representations, vector randomization algorithm, method of fundamental solutions, stochastic projection algorithm, and randomized singular value decomposition are applied. It is shown that the use of stochastic iterative refinement and preconditioning can significantly improve the accuracy and stability of the computations. Simulation results are presented, demonstrating the effectiveness of the proposed methods.

Abstract This study develops explicit algebraic expressions for the single and product moments of order statistics derived from the generalized Bilal (GB) distribution. These expressions facilitate the computation of means, variances and covariances of order statistics for sample sizes up to n = 10 {n=10} with specified parameter values. The derived moments serve as the foundation for constructing the best linear unbiased estimators (BLUEs) and best linear invariant estimators (BLIEs) for the location and scale parameters applicable to both complete and type-II right censored samples. Additionally, the study explores the prediction of unobserved order statistics in type-II right censored samples. The theoretical results are validated through a simulation study, while a real data example highlights their practical utility. These findings establish a robust framework for statistical inference based on order statistics from the GB distribution.

Abstract In this paper, we study the class of first order Periodic Generalized Autoregressive Conditional heteroscedasticity processes ( PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} for short) in which the parameters in the volatility process are allowed to switch between different regimes. First, we establish necessary and sufficient conditions for a PGARCH ⁢ ( 1 , 1 ) {\mathrm{PGARCH}(1,1)} process to have a unique stationary solution (in periodic sense) and for the existence of moments of any order. Next, we are able to estimate the unknown parameters involved in model via the so-called generalized method of moments (GMM). We construct an estimator and establish its asymptotic properties. Specifically, we demonstrate its consistency and asymptotic normality. The GMM estimator in some cases can be more efficient than the least squares estimator (LSE) and the quasi maximum likelihood estimator (QMLE). Some simulation studies are also performed to highlight the impact of our theoretical results.

Abstract The permuted congruential generators is a set of pseudorandom number generators released by Melissa E. O’Neill in 2014. The original technical report outlined several lightweight scrambling techniques designed for the linear congruential generator. Each scrambling technique offered some improvement to the quality of the linear congruential generator. However, the real strength of the scrambling techniques was that they could be combined into multiple overall stronger scramblers. The technical report concludes with the creation of the PCG library, a popular pseudorandom number generation library that implements several generators described in the technical report. Starting from the observation that the paper’s work was narrowly focused on implementing their scrambling techniques for specific linear congruential generators, we explore the permuted congruential generator scrambling techniques and their potential for being applied to other pseudorandom number generators by generalizing the scrambling techniques to work across different pseudorandom number generators.

Abstract In logistic regression models, different patterns of data points in observed data can cause large bias in parameter estimates, especially when separation is present in the observed data. In the frequentist approach, maximum likelihood estimates fail to exist when separation occurs in the observed data. In the Bayesian approach, the existence of posterior means is also affected by the presence of separation depending on the form of prior distributions. In this paper, a non-informative G-prior for Bayesian method is proposed to reduce the bias of the parameter estimation when prior distributions of parameters do not have information and separation is present in the data. In this proposed method, the information from observed data and ideas of a normal regression model are implemented to form the mean and standard deviation of the normal prior distributions. The Markov chain Monte Carlo algorithm is then employed by using Metropolis Hasting algorithm to sample for the target posterior distribution. Results show that estimates from the proposed Bayesian method are more accurate and reliable than from the classical approach when separation is present or is not present in the observed data. Moreover, the proposed Bayesian method can provide better estimated results compared to the default Cauchy prior Bayesian approach when the prior distribution does not have information. The proposed method is also validated by applying it to a case study of MROZ data.

Abstract We show that we can generate nonrecursive 𝑛-bit pseudorandom numbers using a simple algorithm whose essential computation is five times repetition of (𝑛-bit) × \times (𝑛-bit) multiplication and taking out an 𝑛-bit integer from the result of multiplication. The algorithm can be described by 𝛽-transformation T β ⁢ ( X ) = β ⁢ X − ⌊ β ⁢ X ⌋ + 1 , X ∈ [ 1 , 2 ) , β > 1 . T_{\beta}(X)=\beta X-\lfloor\beta X\rfloor+1,\quad X\in[1,2),\,\beta>1. We consider the condition that repetition of 𝛽-transformation generates random numbers, and see why our simple algorithm works well for various values of 𝑛

Abstract In this paper, a stochastic model related to the Rayleigh density function curve is proposed. First, we determined the explicit form of the process by solving the stochastic differential equation by applying the Itô method. Then we determined the probabilistic characteristics such as the density function, the mean and the conditional mean functions. Unlike other processes in the same context, this one allowed us to find the explicit form of the estimators of these parameters by solving the maximum likelihood equations system. In addition, an estimation study on simulated data is carried out in order to validate the efficiency of the estimators proposed by the maximum likelihood methodology. Finally, an application to real data is presented.