Pilar L. Iglesias

A Gibbs sampling scheme to the product partition model: an application to change-point problems

This paper extends previous results for the classical product partition model applied to the identification of multiple change points in the means and variances of time series. Prior distributions for these two parameters and for the probability p that a change takes place at a particular period of time are considered and a new scheme based on Gibbs sampling to estimate the posterior relevances of the model is proposed. The resulting algorithm is applied to the analysis of two Brazilian stock market data. The computational experiments seem to indicate that the algorithm runs fast in common PC-like machines and it may be a useful tool for analyzing change-point problems.

Data analysis using regression models with missing observations and long-memory: an application study

The objective of this work is to propose a statistical methodology to handle regression data exhibiting long memory errors and missing values. This type of data appears very often in many areas, including hydrology and environmental sciences, among others. A generalized linear model is proposed to deal with this problem and an estimation strategy is developed that combines both classical and Bayesian approaches. The estimation methodology proposed is illustrated with an application to air pollution data which shows the impact of the long memory in the statistical inference and of the missing values on the computations. From a Bayesian standpoint, genuine priors are considered for the parameters of the model which are justified within the context of the air pollution model derivation.

Bayesian analysis of the calibration problem under elliptical distributions

Abstract In this paper we discuss calibration problems under dependent and independent elliptical family of distributions. In the dependent case, it is shown that the posterior distribution of the quantity of interest is robust with respect to the distributions in the elliptical family. In particular, the results obtained by Hoadley (1970. J. Amer. Statist. 65, 356–369) showing that the inverse estimator is a Bayes estimator under normal models with a Student-t prior also holds under the dependent elliptical family of distributions. In the independent case, the use of the elliptical family allows the consideration of models which provide protection against possible outliers in the data. The multivariate calibration problem is also considered, where some results given in Brown (1993. Measurement, Regression and Calibration. Oxford University Press, Oxford) are extended. Finally, the results of the paper are applied to a real data problem, showing that the Student-t model can be a valid alternative to normality.

Predictivistic characterizations of multivariate student- t models

De Finetti style theorems characterize models (predictive distributions) as mixtures of the likelihood function and the prior distribution, beginning from some judgment of invariance about observable quantities. The likelihood function generally has its functional form identified from invariance assumptions only. However, we need additional conditions on observable quantities (typically, assumptions on conditional expectations) to identify the prior distribution. In this paper, we consider some well-known invariance assumptions and establish additional conditions on observable quantities in order to obtain a predictivistic characterization of the multivariate and matrix-variate Student-t distributions as well as for the Student-t linear model. As a byproduct, a characterization for the Pearson type II distribution is provided.

BAYESIAN IDENTIFICATION OF OUTLIERS AND CHANGE-POINTS IN MEASUREMENT ERROR MODELS

The problem of outlier and change-point identification has received considerable attention in traditional linear regression models from both, classical and Bayesian standpoints. In contrast, for the case of regression models with measurement errors, also known as error-in-variables models, the corresponding literature is scarce and largely focused on classical solutions for the normal case. The main object of this paper is to propose clustering algorithms for outlier detection and change-point identification in scale mixture of error-in-variables models. We propose an approach based on product partition models (PPMs) which allows one to study clustering for the models under consideration. This includes the change-point problem and outlier detection as special cases. The outlier identification problem is approached by adapting the algorithms developed by Quintana and Iglesias [32] for simple linear regression models. A special algorithm is developed for the change-point problem which can be applied in a more general setup. The methods are illustrated with two applications: (i) outlier identification in a problem involving the relationship between two methods for measuring serum kanamycin in blood samples from babies, and (ii) change-point identification in the relationship between the monthly dollar volume of sales on the Boston Stock Exchange and the combined monthly dollar volumes for the New York and American Stock Exchanges.

A note on Bayesian identification of change points in data sequences

Recent research in mathematical methods for finance suggests that time series for financial data should be studied with non-stationary models and with structural changes that include both jumps and heteroskedasticity (with jumps in variance). It has been recognized that discriminating between variations caused by the continuous motion of Brownian shocks and the genuine discontinuities in the path of the process constitutes a challenge for existing computational procedures. This issue is addressed here, using the product partition model (PPM), for performing such discrimination and the estimation of process jump parameters. Computational implementation aspects of PPM applied to the identification of change points in data sequences are discussed. In particular, we analyze the use of a Gibbs sampling scheme to compute the estimates and show that there is no significant impact of such use on the quality of the results. The influence of the size of the data sequence on the estimates is also of interest, as well as the efficiency of the PPM to correctly identify atypical observations occurring in close instants of time. Extensive Monte Carlo simulations attest to the effectiveness of the Gibbs sampling implementation. An illustrative financial time series example is also presented.

Full predictivistic modeling of stock market data: Application to change point problems

In change point problems in general we should answer three questions: how many changes are there? Where are they? And, what is the distribution of the data within the blocks? In this paper, we develop a new full predictivistic approach for modeling observations within the same block of observation and consider the product partition model (PPM) for treating the change point problem. The PPM brings more flexibility into the change point problem because it considers the number of changes and the instants when the changes occurred as random variables. A full predictivistic characterization of the model can provide a more tractable way to elicit the prior distribution of the parameters of interest, once prior opinions will be required only about observable quantities. We also present an application to the problem of identifying multiple change points in the mean and variance of a stock market return time series.

Characterizations of multivariate spherical distributions inl ∞-norm

The objective of this work is to characterize families of distributions which consist of mixtures of the uniform distributions on the surface of the N-sphere in thel∞-norm. We discuss the characterization through distribution functions and stochastic representations rather than through a measure theoretic approach. Connections with the finite forms of de Finetti-type theorems are considered.

Semiparametric Bayesian measurement error modeling

This work presents a Bayesian semiparametric approach for dealing with regression models where the covariate is measured with error. Given that (1) the error normality assumption is very restrictive, and (2) assuming a specific elliptical distribution for errors (Student-t for example), may be somewhat presumptuous; there is need for more flexible methods, in terms of assuming only symmetry of errors (admitting unknown kurtosis). In this sense, the main advantage of this extended Bayesian approach is the possibility of considering generalizations of the elliptical family of models by using Dirichlet process priors in dependent and independent situations. Conditional posterior distributions are implemented, allowing the use of Markov Chain Monte Carlo (MCMC), to generate the posterior distributions. An interesting result shown is that the Dirichlet process prior is not updated in the case of the dependent elliptical model. Furthermore, an analysis of a real data set is reported to illustrate the usefulness of our approach, in dealing with outliers. Finally, semiparametric proposed models and parametric normal model are compared, graphically with the posterior distribution density of the coefficients.

Comparison between a measurement error model and a linear model without measurement error

The regression of a response variable y on an explanatory variable @x from observations on (y,x), where x is a measurement of @x, is a special case of errors-in-variables model or measurement error model (MEM). In this work we attempt to answer the following question: given the data (y,x) under a MEM, is it possible to not consider the measurement error on the covariable @x in order to use a simpler model? To the best of our knowledge, this problem has not been treated in the Bayesian literature. To answer that question, we compute Bayes factors, the deviance information criterion and the posterior mean of the logarithmic discrepancy. We apply these Bayesian model comparison criteria to two real data sets obtaining interesting results. We conclude that, in order to simplify the MEM, model comparison criteria can be useful to compare structural MEM and a random effect model, but we would also need other statistic tools and take into account the final goal of the model.