Lyle Broemeling

Bayesian inferences related to shifting sequences and two-phase regression

The problem of estimating the switch point in a sequence of independent random variables is studied from a Bayesian viewpoint. Theoretical results and numerical examples are given for the normal sequence and two-phase regression.

Bayesian inferences about a changing sequence of random variables

This study is designed to investigate Bayesian procedures for estimating the time point at which a parameter change occurred in an observed sequence of independent random variables of the regular exponential class. In particular, binomial, exponential, and normal sequences are considered and a generalization to the so-called two-phase regression problem is emphasized. In addition, inference to other parameters of the sequence is made.

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 inference for the variance components in general mixed linear models

The general mixed linear model, containing both the fixed and random effects, is considered. Using gamma priors for the variance components, the conditional posterior distributions of the fixed effects and the variance components, conditional on the random effects, are obtained. Using the normal approximation for the multiple t distribution, approximations are obtained for the posterior distributions of the variance components in infinite series form. The same approximation Is used to obtain closed expressions for the moments of the variance components. An example is considered to illustrate the procedure and a numerical study examines the closeness of the approximations.

Detecting structural change in linear models

Consider a sequence of independent observations which change their marginal distribution at most once somewhere in the sequence and one is not certain where the change has occurred. One would be interested in detecting the change and determining the two distributions which would describe the sequence. On the other hand if no change had occurred, one would want to know the common distribution of the observations. This study develops a Bayesian test for detecting a switch from one linear model to another. The test is based on the marginal posterior mass function of the switch point and the posterior probability of a stable model. This test and an informal sequential procedure of Smith are illustrated with data generated from an unstable linear regression model, which changes the linear relationship between the dependent and independent variables

Inference and prediction with arma processes

An essential ingredient of any time series analysis is the estimation of the model parameters and the forecasting of future observations. This investigation takes a Bayesian approach to the analysis of time series by making inferences of the model parameters from the posterior distribution and forecasting from the predictive distribution. The foundation of the approach is to approximate the condi-tional likelihood by a normal-gamma distribution on the parameter space. The techniques illustrated with many examples of ARMA processes.

Simultaneous inferences for variance ratios of some mixed linear models

Earlier investigations used a one-sided inequality to consltuct confidence regions for the variance ratios or balanced randoiu models. In this study, confidence regions are based on a two-sided generalisation of this inequality and the results are illustrated by estimating the parameters of some elementary random models.

Forecasting future values of changing sequences

Sequences of independent random variables are observed and on the basis of these observations future values of the process are forecast. The Bayesian predictive density of k future observations for normal, exponential, and binomial sequences which change exactly once are analyzed for several cases. It is seen that the Bayesian predictive densities are mixtures of standard probability distributions. For example, with normal sequences the Bayesian predictive density is a mixture of either normal or t-distributions, depending on whether or not the common variance is known. The mixing probabilities are the same as those occurring in the corresponding posterior distribution of the mean(s) of the sequence. The predictive mass function of the number of future successes that will occur in a changing Bernoulli sequence is computed and point and interval predictors are illustrated.

Estimating the parameters of mixed linear models with modal estimators

This paper presents a procedure to estimate the variance components and fixed effects of mixed linear models. The mode of the joint posterior distribution of all the parameters is obtained by an iterative technique. The proposed method is illustrated with one-way and two-fold nested random models. Two numerical examples demonstrate the iterative solution.

The uncertainty of forecasting: models with structural change versus those without changing parameters

The Bayesian predictive density is found for future observations of the unknown dependent variables for a multivariate linear model with a single shift in the regression matrix. A numerical example shows that it is dangerous to predict future observations with an unchanging parameter model when the appropriate model should include structural change.