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We motive and calculate Newton–Cotes quadrature integration variance and compare it directly with Monte Carlo (MC) integration variance. We find an equivalence between deterministic quadrature sampling and random MC sampling by noting that MC random sampling is statistically indistinguishable from a method that uses deterministic sampling on a randomly shuffled (permuted) function. We use this statistical equivalence to regularize the form of permissible Bayesian quadrature integration priors such that they are guaranteed to be objectively comparable with MC. This leads to the proof that simple quadrature methods have expected variances that are less than or equal to their corresponding theoretical MC integration variances. Separately, using Bayesian probability theory, we find that the theoretical standard deviations of the unbiased errors of simple Newton–Cotes composite quadrature integrations improve over their worst case errors by an extra dimension independent factor ∝N-12{\propto N^{-\frac{1}{2}}}. This dimension independent factor is validated in our simulations.

Ecological studies and epidemiology need to use group averaged data to make inferences about individual patterns. However, using correlations based on averages to estimate correlations of individual scores is subject to an “ecological fallacy”. The purpose of this article is to create distributions of Pearson R correlation values computed from grouped averaged or aggregate data using Monte Carlo simulations and random sampling. We show that, as the group size increases, the distributions can be approximated by a generalized hypergeometric distribution. The expectation of the constructed distribution slightly underestimates the individual Pearson R value, but the difference becomes smaller as the number of groups increases. The approximate normal distribution resulting from Fisher’s transformation can be used to build confidence intervals to approximate the Pearson R value based on individual scores from the Pearson R value based on the aggregated scores.

The aim of this paper is to show the approximation of Euler–Maruyama Xtn{X_{t}^{n}} for one-dimensional stochastic differential equations involving the maximum process. In addition to that it proves the strong convergence of the Euler–Maruyama whose both drift and diffusion coefficients are Lipschitz. After that, it generalizes to the non-Lipschitz case.

Bayesian approaches have been used in the literature to estimate the parameters for joint models of longitudinal and time-to-event data. The main aim of this paper is to analyze the impact of prior distributions on estimating parameters in a proposed fully Bayesian analysis setting for the penalized spline joint models. To achieve this aim, the joint posterior distribution of parameters in survival and longitudinal submodels is presented. The Markov chain Monte Carlo (MCMC) algorithm is then proposed, which consists of the Gibbs sampler (GS) and Metropolis Hastings (MH) algorithms to sample for the target conditional posterior distributions. The prior sensitivity analysis for the baseline hazard rate and association parameters is performed through simulation studies and a case study.

A Bayesian procedure for bandwidth selection in kernel circular density estimation is investigated, when the Markov chain Monte Carlo (MCMC) sampling algorithm is utilized for Bayes estimates. Under the quadratic and entropy loss functions, the proposed method is evaluated through a simulation study and real data sets, which were already discussed in the literature. The proposed Bayesian approach is very competitive in comparison with the existing classical global methods, namely plug-in and cross-validation techniques.