A bayesian approach to the comparison of two means

A hierarchical Bayesian approach to the problem of comparison of two means is considered. Hypothesis testing, ranking and selection, and estimation (after selection) are treated. Under the hypothesis that two means are different, it is desired to select the population which has the larger mean. Expressions for the ranking probability of each mean being the larger and the corresponding estimate of each mean are given. For certain priors, it is possible to express the quantities of interest in closed form. A simulation study has been done to compare mean square errors of a hierarchical Bayesian estimator and some of the existing estimators of the selected mean. The case of comparing two means in the presence of block effects has also been considered and an example is presented to illustrate the methodology.     

Inference for post-change mean by a CUSUM procedure

In this paper, the inference problem for the post-change mean is considered after a change is detected by a CUSUM process in a sequence of independent normal variables. The change-point is estimated as the maximum likelihood estimate at the reference value and the post-change mean is estimated as the sample mean after the change-point estimate. By assuming the change-point is large and the monitoring limit approaches infinity, the first-order bias of the post-change mean estimate and a corrected asymptotic normal pivot are derived conditioning on that a change is detected. Local approximations for small reference value and post-change mean are obtained for numerical evaluation.     

A Bayesian multiple structural change regression model with autocorrelated errors

This paper develops a new Bayesian approach to change-point modeling that allows the number of change-points in the observed autocorrelated times series to be unknown. The model we develop assumes that the number of change-points have a truncated Poisson distribution. A genetic algorithm is used to estimate a change-point model, which allows for structural changes with autocorrelated errors. We focus considerable attention on the construction of autocorrelated structure for each regime and for the parameters that characterize each regime. Our techniques are found to work well in the simulation with a few change-points. An empirical analysis is provided involving the annual flow of the Nile River and the monthly total energy production in South Korea to lead good estimates for structural change-points.     

Variance components estimation for the balanced random effects model under mixed prior distributions

For the balanced random effects models, when the variance components are correlated either naturally or through common prior structures, by assuming a mixed prior distribution for the variance components, we propose some new Bayesian estimators. To contrast and compare the new estimators with the minimum variance unbiased (MVUE) and restricted maximum likelihood estimators (RMLE), some simulation studies are also carried out. It turns out that the proposed estimators have smaller mean squared errors than the MVUE and RMLE.     

C337. Misspecified location and scale. 11. independence of and s2

A hierarchical Bayesian approach to the problem of estimating the largest normal mean is considered. Calculation of the posterior mean and the posterior variance involves, at worst, 3-dimensional numerical integration, for which an efficient Monte Carlo method of evaluation is given. An example is presented to illustrate the methodology. In the two populations case, computation of the posterior estimates can be substantially simplified and in special cases can actually be performed using closed form solutions. A simulation study has been done to compare mean square errors of some hierarchical Bayesian estimators that are expressed in closed forms and several existing estimators of the larger mean.     

A Bayesian wavelet approach to estimation of a change-point in a nonlinear multivariate time series

ABSTRACT

We propose a semiparametric approach to estimate the existence and location of a statistical change-point to a nonlinear multivariate time series contaminated with an additive noise component. In particular, we consider a p-dimensional stochastic process of independent multivariate normal observations where the mean function varies smoothly except at a single change-point. Our approach involves conducting a Bayesian analysis on the empirical detail coefficients of the original time series after a wavelet transform. If the mean function of our time series can be expressed as a multivariate step function, we find our Bayesian-wavelet method performs comparably with classical parametric methods such as maximum likelihood estimation. The advantage of our multivariate change-point method is seen in how it applies to a much larger class of mean functions that require only general smoothness conditions.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Bayesian multiple change-point estimation for exponential distribution with truncated and censored data

This paper considers the multiple change-point estimation for exponential distribution with truncated and censored data by Gibbs sampling. After all the missing data of interest is filled in by some sampling methods such as rejection sampling method, the complete-data likelihood function is obtained. The full conditional distributions of all parameters are discussed. The means of Gibbs samples are taken as Bayesian estimations of the parameters. The implementation steps of Gibbs sampling are introduced in detail. Finally random simulation test is developed, and the results show that Bayesian estimations are fairly accurate.     

Block permutation principles for the change analysis of dependent data

We study an AMOC model with an abrupt change in the mean and dependent errors that form a linear process. Different kinds of statistics are considered, such as maximum-type statistics (particularly different CUSUM procedures) or sum-type statistics. Approximations of the critical values for change-point tests are obtained through permutation methods. The theoretical results show that the original test statistics and their corresponding block permutation counterparts follow the same distributional asymptotics. The main step in the proof is to obtain limit theorems for the corresponding rank statistics and then use laws of large numbers to obtain the permutation asymptotics conditionally on the given data.     

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.     

Change Point Detection in The Skew-Normal Model Parameters

Bayesian inference under the skew-normal family of distributions is discussed using an arbitrary proper prior for the skewness parameter. In particular, we review some results when a skew-normal prior distribution is considered. Considering this particular prior, we provide a stochastic representation of the posterior of the skewness parameter. Moreover, we obtain analytical expressions for the posterior mean and variance of the skewness parameter. The ultimate goal is to consider these results to one change point identification in the parameters of the location-scale skew-normal model. Some Latin American emerging market datasets are used to illustrate the methodology developed in this work.     

Estimation in multi-path change-point problems

The current literature deals with the change-point problem only in the context of the obser¬vation of a single sequence. In this paper, inference will be based on the observation of TV sequences of random variables, each sequence containing one change-point. This extension allows the effective use of bootstrap and empirical Bayes methods, both of which are not feasible in the single-path context. Two classes of these “multi-path” change-point problems are considered. If the change-point is assumed to occur at the the same position in each sequence, then the terminology “fixed-tau multi-path change-point” will be used. In other cases, one may expect the change-point to occur at random positions in each sequence, according to some distribution, a “random-tau multi-path change-point” problem. Examples and simulations are given.     

On robust Bayesian estimation under some asymmetric and bounded loss function

The problem of Bayesian and robust Bayesian estimation with some bounded and asymmetric loss function ABL is considered for various models. The prior distribution is not exactly specified and covers the conjugate family of prior distributions. The posterior regret, most robust and conditional Γ-minimax estimators are constructed and a preliminary comparison with square-error loss and LINEX loss is presented.     

Statistical Properties of Model-Based Signal Extraction Diagnostic Tests

The problem of choosing the loss function in the Bayesian problem of many hypotheses testing is considered. It is shown that linear and quadratic loss functions are the most-used ones. For any kind of loss function, the risk function in the Bayesian problem of many hypotheses testing contains the errors of two kinds. The Bayesian decision rule minimizes the total effect of these errors. The share of each of them in the optimal value of risk function is unknown. When solving many important problems, the results caused by different errors significantly differ from each other. Therefore, it is necessary to guarantee the limitation on the most undesirable kind of these errors and to minimize the errors of the second kind. For solving these problems, this article are states and solves conditional Bayesian tasks of testing many hypotheses. The results of sensitivity analysis of the classical and conditional Bayesian problems are given and their advantages and drawbacks are considered.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

Bayesian D-optimal design for logistic regression model with exponential distribution for random intercept

ABSTRACT

Nowadays, generalized linear models have many applications. Some of these models which have more applications in the real world are the models with random effects; that is, some of the unknown parameters are considered random variables. In this article, this situation is considered in logistic regression models with a random intercept having exponential distribution. The aim is to obtain the Bayesian D-optimal design; thus, the method is to maximize the Bayesian D-optimal criterion. For the model was considered here, this criterion is a function of the quasi-information matrix that depends on the unknown parameters of the model. In the Bayesian D-optimal criterion, the expectation is acquired in respect of the prior distributions that are considered for the unknown parameters. Thus, it will only be a function of experimental settings (support points) and their weights. The prior distribution of the fixed parameters is considered uniform and normal. The Bayesian D-optimal design is finally calculated numerically by R3.1.1 software.     

Objective Bayesian analysis of spatial data with measurement error

The author shows how geostatistical data that contain measurement errors can be analyzed objectively by a Bayesian approach using Gaussian random fields. He proposes a reference prior and two versions of Jeffreys' prior for the model parameters. He studies the propriety and the existence of moments for the resulting posteriors. He also establishes the existence of the mean and variance of the predictive distributions based on these default priors. His reference prior derives from a representation of the integrated likelihood that is particularly convenient for computation and analysis. He further shows that these default priors are not very sensitive to some aspects of the design and model, and that they have good frequentist properties. Finally, he uses a data set of carbon/nitrogen ratios from an agricultural field to illustrate his approach.     

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.     

Asymptotic study of the change-point mle in multivariate Gaussian families under contiguous alternatives

The problem of estimating an unknown change-point in the mean vector or covariance matrix of a sequence of independent multivariate Gaussian random variables is considered. Adapting the estimation methodology that Hinkley pursued for the case of abrupt changes, we develop theory for deriving the asymptotic distribution of the maximum likelihood estimator of the change-point when the amount of change is a function of the sample size and goes to zero in a smooth fashion as the sample size goes to infinity, yielding a contiguous change-point model. Simulations have been performed to illustrate the closeness of the asymptotic distribution with the empirical distribution, and to evaluate its robustness to departures from normality for reasonable sample sizes as well as parameter changes. Finally, we apply the methodology to estimate the change-point in the daily log-returns data of BLS (BellSouth) and VZ (Verizon) from NYSE.     

A note on predictive inference for multivariate elliptically contoured distributions

The prediction problem is considered for the multivariate regression model with an elliptically contoured error distribution. We show that the predictive distribution under elliptical errors assumption is the same as that obtained under normally distributed error in both the Bayesian approach using an im-proper prior and the classical approach. This gives inference robustness with respect to departures from the reference case of independent sampling from the normal distribution.