On ar(1) processes with exponential white noise

We consider the autoregressive model X_t= bX_t-1= Y_twhere 0 ≤ b < 1 and Y_tare independent random variables with an exponential distribution. The moments of the stationary distribution of X_tare calculated and the distribution of an approximation to the maximum likelihood estimator for b is derived. The result is used for a construction of a confidence interval for b.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

Bayesian assessment of assumptions of regression analysis

Many statistical procedures are based on the models which specify the conditions under which the data are generated. Many applications of linear regression, for example, assume that:(i) the observations are independent; (ii) the errors in the observations are identically distributed; (iii) each error has a normal distribution with mean zero and unknown variance σ²> 0. Previous works have examined individual departures from these assumptions. Here we examine composite departures. It is assumed that the error distribution in a linear model is power-exponential and that the observations are generated via a first order autoregressive model with the possibility of spurious observations. The consequences are illustrated via an example.     

Exact predictors for a generalized ar(1) process with an ar(1) parameter

The generalized AR(1) process y _t = a _t y _t-1+ v _t is considered, where the parameter a _t follows the AR(1) process a _t = Ga _t-1+ w _t.Assuming that V _t and w _t are Gaussian and independent, the first six exact predictors for future values of y _t are derived. These exact predictors are compared with Box-Jenkins -type approximations. MACSYMA, a computer algebra program, is utilized in the derivation of the predictors.     

Bayesian analysis of circular distributions based on non-negative trigonometric sums

ABSTRACT

Fernández-Durán [Circular distributions based on nonnegative trigonometric sums. Biometrics. 2004;60:499–503] developed a new family of circular distributions based on non-negative trigonometric sums that is suitable for modelling data sets that present skewness and/or multimodality. In this paper, a Bayesian approach to deriving estimates of the unknown parameters of this family of distributions is presented. Because the parameter space is the surface of a hypersphere and the dimension of the hypersphere is an unknown parameter of the distribution, the Bayesian inference must be based on transdimensional Markov Chain Monte Carlo (MCMC) algorithms to obtain samples from the high-dimensional posterior distribution. The MCMC algorithm explores the parameter space by moving along great circles on the surface of the hypersphere. The methodology is illustrated with real and simulated data sets.     

A conjugate family for ar(1) processes with exponential errors

In this article estimation of autoregressive processes AR(1) with exponential errors, denoted by ARE(1), is considered from a Bayesian perspective. For these processes a new family of conjugate distributions, denoted by GBTP, is shown to exist which follows for recursive estimation of the parameters in the model. Further extensions of the model are also considered.     

Likelihood ratio type unit root tests for ar(1)models with nonconsecutive observations

For the nonconsecutively observed or missing data situation likelihood ratio type unit root tests in AR(1)models containing an intercept or both an intercept and a time trend are proposed and are shown to have the same limiting distributions as the likelihood ratio tests for the complete data case as tabulated by Dickey and Fuller(1981). Some simulation results on our tests in finite samples under A–B sampling schemes are also presented.     

Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

Geometric stick-breaking processes for continuous-time Bayesian nonparametric modeling

We propose a new class of time dependent random probability measures and show how this can be used for Bayesian nonparametric inference in continuous time. By means of a nonparametric hierarchical model we define a random process with geometric stick-breaking representation and dependence structure induced via a one dimensional diffusion process of Wright-Fisher type. The sequence is shown to be a strongly stationary measure-valued process with continuous sample paths which, despite the simplicity of the weights structure, can be used for inferential purposes on the trajectory of a discretely observed continuous-time phenomenon. A simple estimation procedure is presented and illustrated with simulated and real financial data.     

Bayesian computation for Log-Gaussian Cox processes: a comparative analysis of methods

The Log-Gaussian Cox process is a commonly used model for the analysis of spatial point pattern data. Fitting this model is difficult because of its doubly stochastic property, that is, it is a hierarchical combination of a Poisson process at the first level and a Gaussian process at the second level. Various methods have been proposed to estimate such a process, including traditional likelihood-based approaches as well as Bayesian methods. We focus here on Bayesian methods and several approaches that have been considered for model fitting within this framework, including Hamiltonian Monte Carlo, the Integrated nested Laplace approximation, and Variational Bayes. We consider these approaches and make comparisons with respect to statistical and computational efficiency. These comparisons are made through several simulation studies as well as through two applications, the first examining ecological data and the second involving neuroimaging data.     

Bayesian inference for smooth transition autoregressive (STAR) model: A prior sensitivity analysis

The main aim of this paper is to perform sensitivity analysis to the specification of prior distributions in a Bayesian analysis setting of STAR models. To achieve this aim, the joint posterior distribution of model order, coefficient, and implicit parameters in the logistic STAR model is first being presented. The conditional posterior distributions are then shown, followed by the design of a posterior simulator using a combination of Metropolis-Hastings, Gibbs Sampler, RJMCMC, and Multiple Try Metropolis algorithms, respectively. Following this, simulation studies and a case study on the prior sensitivity for the implicit parameters are being detailed at the end.     

Bayesian analysis of spatial point processes in the neighbourhood of Voronoi networks

A model for an inhomogeneous Poisson process with high intensity near the edges of a Voronoi tessellation in 2D or 3D is proposed. The model is analysed in a Bayesian setting with priors on nuclei of the Voronoi tessellation and other model parameters. An MCMC algorithm is constructed to sample from the posterior, which contains information about the unobserved Voronoi tessellation and the model parameters. A major element of the MCMC algorithm is the reconstruction of the Voronoi tessellation after a proposed local change of the tessellation. A simulation study and examples of applications from biology (animal territories) and material science (alumina grain structure) are presented.     

A Bayesian approach for non-homogeneous gamma degradation processes

A Bayesian approach based on the Markov Chain Monte Carlo technique is proposed for the non-homogeneous gamma process with power-law shape function. Vague and informative priors, formalized on some quantities having a “physical” meaning, are provided. Point and interval estimation of process parameters and some functions thereof are developed, as well as prediction on some observable quantities that are useful in defining the maintenance strategy is proposed. Some useful approximations are derived for the conditional and unconditional mean and median of the residual life to reduce computational time. Finally, the proposed approach is applied to a real dataset.     

Strict stationarity of ar(p) processes generated by nonlinear random functions with additive perturbations

Let {X_n} be a generalized autoregressive process of order ρ defined by X_n=Φ_n(X_n-ρ,…,X_n-1)-η_m, where {φ_n} is a sequence of i.i.d. random maps taking values on H, and {η_n} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on R_P to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {X_n}.     

Bayesian analysis of hierarchical linear mixed modeling using the multivariate t distribution

This article presents a fully Bayesian approach to modeling incomplete longitudinal data using the t linear mixed model with AR(p) dependence. Markov chain Monte Carlo (MCMC) techniques are implemented for computing posterior distributions of parameters. To facilitate the computation, two types of auxiliary indicator matrices are incorporated into the model. Meanwhile, the constraints on the parameter space arising from the stationarity conditions for the autoregressive parameters are handled by a reparametrization scheme. Bayesian predictive inferences for the future vector are also investigated. An application is illustrated through a real example from a multiple sclerosis clinical trial.     

A Bayesian analysis of a change in the parameters of autoregressive time series

In this article, we consider a Bayesian analysis of a possible change in the parameters of autoregressive time series of known order p, AR(p). An unconditional Bayesian test based on highest posterior density (HPD) credible sets is determined. The test is useful to detect a change in any one of the parameters separately. Using the Gibbs sampler algorithm, we approximate the posterior densities of the change point and other parameters to calculate the p-values that define our test.     

Bayesian analysis of a linear mixed model with AR(p) errors via MCMC

We develop Bayesian procedures to make inference about parameters of a statistical design with autocorrelated error terms. Modelling treatment effects can be complex in the presence of other factors such as time; for example in longitudinal data. In this paper, Markov chain Monte Carlo methods (MCMC), the Metropolis-Hastings algorithm and Gibbs sampler are used to facilitate the Bayesian analysis of real life data when the error structure can be expressed as an autoregressive model of order p. We illustrate our analysis with real data.