Parameter changes in a regression model with autocorrelated errors

This study is a Bayesian analysis of a regression model with autocorrelated errors which exhibits one change in the regression parameters and where the autocorrelation parameter is unknown

Using a normal-gamma prior for all the parameters except the shift point which has a uniform distribution, the marginal posterior distribution of the regression parameters, the shift point and the precision of the errors is found. It is important to know where the shift occurred thus the main emphasis is with the posterior distribution of the shift point

A numerical study assesses the effect of the values of the shift point and the magnitude of the shift on the posterior distribution of the shift point. The posterior distribution of the shift point is more sensitive to change, which occurs in the middle of the observations than to one which occurs at an extreme of the data.     

Bayesian forecasting with changing linear models

This investigation considers a general linear model which changes parameters exactly once during the observation period. Assuming all the parameters are unknown and a proper prior distribution, the Bayesian predictive distribution of the future observations is derived.

It is shown that the predictive distribution is a mixture of multivariate t distributions and that the mixing distribution is the marginal posterior mass function of the change point parameter.     

Bayesian analysis of linear models exhibiting changes in mean and precisionat an unknown time point

This paper analyses a linear model in which both the mean and the precision change exactly once at an unknown point in time. Posterior distributions are found for the unknown time point at which the changes occurred and for the ratio of the precisions. The Bayesian predictive distribution of k future observations is also derived. It is shown that the unconditional posterior distribution of the ratio of precisions is a mixture of F-type distributions and the predictive distribution is a mixture of multivariate t distributions.     

Testing for a change lit the regression matrix of a multivariate linear model

A Bayesian test procedure Is developed to test; the null hypothesis of no change In the regression matrix of a multivariate lin¬ear model against the alternative hypothesis of exactly one change The resulting test is based on the marginal posterior distribution of the change point; To illustrate the test procedure a numerical example using a bivariate regression model is considered.     

Structural changes in multivariate regression models

This study generalizes the work of chin choy and Broemeling (1980) who investigated the change in the regression parameters of univariate linear models.

The marginal posterior distributions of the change point, the regression matrices,and the precision matrix are found with the use of a proper multivariate normal-Wishart distribution for the parameters of the model.

A numerical study is undertaken in order to gain some insight into the effect that changes in sample size and certain parameter values have on these marginal posterior distributions.     

Retrospective change detection in categorical time series

The aim of this paper is to propose methods of detecting change in the coefficients of a multinomial logistic regression model for categorical time series offline. The alternatives to the null hypothesis of stationarity can be either the hypothesis that it is not true, or that there is a temporary change in the sequence. We use the efficient score vector of the partial likelihood function. This has several advantages. First, the alternative value of the parameter does not have to be estimated; hence, we have a procedure that has a simple structure with only one parameter estimation using all available observations. This is in contrast with the generalized likelihood ratio-based change point tests. The efficient score vector is used in various ways. As a vector, its components correspond to the different components of the multinomial logistic regression model’s parameter vector. Using its quadratic form a test can be defined, where the presence of a change in any or all parameters is tested for. If there are too many parameters one can test for any subset while treating the rest as nuisance parameters. Our motivating example is a DNA sequence of four categories, and our test result shows that in the published data the distribution of the four categories is not stationary.     

Parametric and semiparametric hypotheses in the linear model

The independent additive errors linear model consists of a structure for the mean and a separate structure for the error distribution. The error structure may be parametric or it may be semiparametric. Under alternative values of the mean structure, the best fitting additive errors model has an error distribution which can be represented as the convolution of the actual error distribution and the marginal distribution of a misspecification term. The model misspecification term results from the covariates' distribution. Conditions are developed to distinguish when the semiparametric model yields sharper inference than the parametric model and vice versa. The main conditions concern the actual error distribution and the covariates' distribution. The theoretical results explain a paradoxical finding in semiparametric Bayesian modelling, where the posterior distribution under a semiparametric model is found to be more concentrated than is the posterior distribution under a corresponding parametric model. The paradox is illustrated on a set of allometric data. The Canadian Journal of Statistics 39: 165–180; 2011 ©2011 Statistical Society of Canada     

Bayesian analysis of outlier problems using divergence measures

A Bayesian approach is presented for detecting influential observations using general divergence measures on the posterior distributions. A sampling-based approach using a Gibbs or Metropolis-within-Gibbs method is used to compute the posterior divergence measures. Four specific measures are proposed, which convey the effects of a single observation or covariate on the posterior. The technique is applied to a generalized linear model with binary response data, an overdispersed model and a nonlinear model. An asymptotic approximation using Laplace method to obtain the posterior divergence is also briefly discussed.     

Retrospective Parametric Tests for Homogeneity of Data

The problem considered in this study is that of detecting a change in the unknown parameters of known distribution on the basis of a finite sequence of independent observations, assuming that if a change in the parameters of a distribution has occurred then it is unique. We examine various methods that have been suggested for this problem and suggest a uniform approach, based on likelihood ratio analysis. For tests derived this way we present approximations for levels of significance based on asymptotic analyses. The suggested tests meet the needs of specific problems (such as a one-sided alternative), for which general parametric case solutions have not been suggested explicitly before. We also find that rate of convergence of our asymptotics is fast, and provide accurate results for a level of significance of the suggested tests for sample sizes commonly observed in practice.     

Data reconciliation of nonnormal observations with nonlinear constraints

This paper presents a new method for the reconciliation of data described by arbitrary continuous probability distributions, with the focus on nonlinear constraints. The main idea, already applied to linear constraints in a previous paper, is to restrict the joint prior probability distribution of the observed variables with model constraints to get a joint posterior probability distribution. Because in general the posterior probability density function cannot be calculated analytically, it is shown that it has decisive advantages to sample from the posterior distribution by a Markov chain Monte Carlo (MCMC) method. From the resulting sample of observed and unobserved variables various characteristics of the posterior distribution can be estimated, such as the mean, the full covariance matrix, marginal posterior densities, as well as marginal moments, quantiles, and HPD intervals. The procedure is illustrated by examples from material flow analysis and chemical engineering.     

A note on the effect of simple equicorrelation in detecting a spurious multivariate observation

This paper examines the robustness of the multivariate version of Grubs' (1950) procedure for detecting an outlier in a sample of n independent observations against equicorrelation of the observations. It is shown that the robustness of the univariate test to equicorrelation extends to the multivariate test in that the distribution of the maximum squared radii-test for a multivariate oulier in identical for both the independent and siaply equicorrelated data models.     

A Simulation-based Goodness-of-fit Test for Random Effects in Generalized Linear Mixed Models

Abstract.  The goodness-of-fit of the distribution of random effects in a generalized linear mixed model is assessed using a conditional simulation of the random effects conditional on the observations. Provided that the specified joint model for random effects and observations is correct, the marginal distribution of the simulated random effects coincides with the assumed random effects distribution. In practice, the specified model depends on some unknown parameter which is replaced by an estimate. We obtain a correction for this by deriving the asymptotic distribution of the empirical distribution function obtained from the conditional sample of the random effects. The approach is illustrated by simulation studies and data examples.     

A binary control chart to detect small jumps

The classic N p-chart gives a signal if the number of successes in a sequence of independent binary variables exceeds a control limit. Motivated by engineering applications in industrial image processing and, to some extent, financial statistics, we study a simple modification of this chart, which uses only the most recent observations. Our aim is to construct a control chart for detecting a shift of an unknown size, allowing for an unknown distribution of the error terms. Simulation studies indicate that the proposed chart is superior in terms of out-of-control average run length, when one is interested in the detection of very small shifts. We provide a (functional) central limit theorem under a change-point model with local alternatives, which explains that unexpected and interesting behaviour. Since real observations are often not independent, the question arises whether these results still hold true for the dependent case. Indeed, our asymptotic results work under the fairly general condition that the observations form a martingale difference array. This enlarges the applicability of our results considerably, first, to a large class of time series models, and, second, to locally dependent image data, as we demonstrate by an example.     

Monte Carlo approximation of likelihood function in spatial GLMMs through an empirical Bayes method

In spatial generalized linear mixed models (SGLMMs), statistical inference encounters problems, since random effects in the model imply high-dimensional integrals to calculate the marginal likelihood function. In this article, we temporarily treat parameters as random variables and express the marginal likelihood function as a posterior expectation. Hence, the marginal likelihood function is approximated using the obtained samples from the posterior density of the latent variables and parameters given the data. However, in this setting, misspecification of prior distribution of correlation function parameter and problems associated with convergence of Markov chain Monte Carlo (MCMC) methods could have an unpleasant influence on the likelihood approximation. To avoid these challenges, we utilize an empirical Bayes approach to estimate prior hyperparameters. We also use a computationally efficient hybrid algorithm by combining inverse Bayes formula (IBF) and Gibbs sampler procedures. A simulation study is conducted to assess the performance of our method. Finally, we illustrate the method applying a dataset of standard penetration test of soil in an area in south of Iran.     

Multivariate Bayesian variable selection and prediction

The multivariate regression model is considered with p regressors. A latent vector with p binary entries serves to identify one of two types of regression coefficients: those close to 0 and those not. Specializing our general distributional setting to the linear model with Gaussian errors and using natural conjugate prior distributions, we derive the marginal posterior distribution of the binary latent vector. Fast algorithms aid its direct computation, and in high dimensions these are supplemented by a Markov chain Monte Carlo approach to sampling from the known posterior distribution. Problems with hundreds of regressor variables become quite feasible. We give a simple method of assigning the hyperparameters of the prior distribution. The posterior predictive distribution is derived and the approach illustrated on compositional analysis of data involving three sugars with 160 near infrared absorbances as regressors.     

Some recent developments for regression analysis of multivariate failure time data

Cox's seminal 1972 paper on regression methods for possibly censored failure time data popularized the use of time to an event as a primary response in prospective studies. But one key assumption of this and other regression methods is that observations are independent of one another. In many problems, failure times are clustered into small groups where outcomes within a group are correlated. Examples include failure times for two eyes from one person or for members of the same family.This paper presents a survey of models for multivariate failure time data. Two distinct classes of models are considered: frailty and marginal models. In a frailty model, the correlation is assumed to derive from latent variables (frailties) common to observations from the same cluster. Regression models are formulated for the conditional failure time distribution given the frailties. Alternatively, marginal models describe the marginal failure time distribution of each response while separately modelling the association among responses from the same cluster.We focus on recent extensions of the proportional hazards model for multivariate failure time data. Model formulation, parameter interpretation and estimation procedures are considered.     

Influential observations in transformations: A bayesian approach

This paper presents a bayesian approach to the problem of detecting influential observations when estimating the Box-Cox transformation. The influence of a group I={i₁, …,i_n} of observations is measured by means of the Kullback-Leibler distance between the marginal posterior; distributions for the transformation parameter which are computed, respectively, without and with the cases indexed by I. A measure is proposed and its properties and relationship to other diagnostic methods are studied.     

Detecting change points and monitoring biomedical data

Bayesian and likelihood approaches to on-line detecting change points in time series are discussed and applied to analyze biomedical data. Using a linear dynamic model, the Bayesian analysis outputs the conditional posterior probability of a change at time t ? 1, given the data up to time t and the status of changes occurred before time t ? 1. The likelihood method is based on a change-point regression model and tests whether there is no change-point.     

A NOTE ON A SEQUENTIAL GROUP t-TEST

A sequential probability ratio test (SPET) of the mean of a normal distribution with unknown variance, based on an independent sequence of groups of observations, is investigated and its efficiency compared with that of the WAGE sequential t-test, which is based on an invariantly sufficient sequence of test statistics.