Brazilian Journal of Probability and Statistics Articles (Project Euclid)

An estimation method for latent traits and population parameters in Nominal Response Model

Thu, 05 Aug 2010 15:41 EDT

Caio L. N. Azevedo, Dalton F. Andrade

Source: Braz. J. Probab. Stat., Volume 24, Number 3, 415--433.

Abstract:
The nominal response model (NRM) was proposed by Bock [ Psychometrika 37 (1972) 29–51] in order to improve the latent trait (ability) estimation in multiple choice tests with nominal items. When the item parameters are known, expectation a posteriori or maximum a posteriori methods are commonly employed to estimate the latent traits, considering a standard symmetric normal distribution as the latent traits prior density. However, when this item set is presented to a new group of examinees, it is not only necessary to estimate their latent traits but also the population parameters of this group. This article has two main purposes: first, to develop a Monte Carlo Markov Chain algorithm to estimate both latent traits and population parameters concurrently. This algorithm comprises the Metropolis–Hastings within Gibbs sampling algorithm (MHWGS) proposed by Patz and Junker [ Journal of Educational and Behavioral Statistics 24 (1999b) 346–366]. Second, to compare, in the latent trait recovering, the performance of this method with three other methods: maximum likelihood, expectation a posteriori and maximum a posteriori. The comparisons were performed by varying the total number of items (NI), the number of categories and the values of the mean and the variance of the latent trait distribution. The results showed that MHWGS outperforms the other methods concerning the latent traits estimation as well as it recoveries properly the population parameters. Furthermore, we found that NI accounts for the highest percentage of the variability in the accuracy of latent trait estimation.

Prediction-based estimating functions: Review and new developments

Sun, 21 Aug 2011 20:37 EDT

Michael Sørensen

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 362--391.

Abstract:
The general theory of prediction-based estimating functions for stochastic process models is reviewed and extended. Particular attention is given to optimal estimation, asymptotic theory and Gaussian processes. Several examples of applications are presented. In particular, partial observation of a system of stochastic differential equations is discussed. This includes diffusions observed with measurement errors, integrated diffusions, stochastic volatility models, and hypoelliptic stochastic differential equations. The Pearson diffusions, for which explicit optimal prediction-based estimating functions can be found, are briefly presented.

Local linear suppression for wireless sensor network data

Sun, 21 Aug 2011 20:37 EDT

Kristian Lum, Alan E. Gelfand

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 392--405.

Abstract:
With wireless sensor networks, preserving battery life is critical. For such sensors, data collection is relatively cheap while data transmission is relatively expensive. For such networks in ecological settings, certain processes are sufficiently predictable so that transmission of data at a particular time can be suppressed if it does not differ from what is expected at that time. That is, there will not be much loss of information with regard to inference. More precisely, there is a presumed model to explain the measurements collected at the sensors, which provides insight into what is expected at a given node, at a given time. Under the suppression, inference objectives include both estimation of the process parameters as well as reconstruction of the entire time series at each of the nodes. In this paper, we build on the existing literature that has offered ways in which one can use suppression in wireless sensor networks to limit the number of transmissions. We introduce a new, computationally cheap, locally linear suppression scheme based upon process knowledge and compare it to the commonly used “constant” suppression scheme. Maintaining the same suppression threshold, we demonstrate decreased transmission rates under the new scheme while producing comparable posterior inference relative to constant suppression scheme. That is, the untransmitted readings are bounded to within an interval of the same length under both schemes, but the linear suppression scheme will transmit less data. We implement this scheme for a synthetic dataset produced under the assumption of a diffusion model and show that even under high suppression rates, we are able to recover simulation parameters. We also implement linear suppression on data collected from a real wireless sensor network that measures the amount of light filtering through the forest canopy at a set of locations in the Duke Forest. We show that the in-sample predictive sum of squared errors from the suppressed data is only a bit larger than that from the full dataset.

Hierarchical wavelet modelling of environmental sensor data

Sun, 21 Aug 2011 20:37 EDT

Yann Ruffieux, A. C. Davison

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 406--420.

Abstract:
Motivated by the need to smooth and to summarize multiple simultaneous time series arising from networks of environmental monitors, we propose a hierarchical wavelet model for which estimation of hyperparameters can be performed by marginal maximum likelihood. The result is an empirical Bayes thresholding procedure whose results improve on those of wavethresh in terms of mean square error. We apply the approach to data from the SensorScope environmental modelling system, and briefly discuss issues that arise concerning variance estimation in this context.

Modelling particles moving in a potential field with pairwise interactions and an application

Sun, 21 Aug 2011 20:37 EDT

D. R. Brillinger, H. K. Preisler, M. J. Wisdom

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 421--436.

Abstract:
Motions of particles in fields characterized by real-valued potential functions, are considered. Three particular expressions for potential functions are studied. One, U , depends on the i th particle’s location, r i ( t ) at times t i . A second, V , depends on particle i ’s vector distances from others, r i ( t )− r j ( t ). This function introduces pairwise interactions. A third, W , depends on the Euclidian distances, ‖ r i ( t )− r j ( t )‖ between particles at the same times, t . The functions are motivated by classical mechanics. Taking the gradient of the potential function, and adding a Brownian term one, obtains the stochastic equation of motion d r i =−∇ U ( r i ) dt −∑ j ≠ i ∇ V ( r i − r j ) dt + σ d B i in the case that there are additive components U and V . The ∇ denotes the gradient operator. Under conditions the process will be Markov and a diffusion. By estimating U and V at the same time one could address the question of whether both components have an effect and, if yes, how, and in the case of a single particle, one can ask is the motion purely random? An empirical example is presented based on data describing the motion of elk ( Cervus elaphus ) in a United States Forest Service reserve.

On improved estimation for importance sampling

Sun, 21 Aug 2011 20:37 EDT

David Firth

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 437--443.

Abstract:
The standard estimator used in conjunction with importance sampling in Monte Carlo integration is unbiased but inefficient. An alternative estimator is discussed, based on the idea of a difference estimator, which is asymptotically optimal. The improved estimator uses the importance weight as a control variate, as previously studied by Hesterberg (Ph.D. Dissertation, Stanford University (1988); Technometrics 37 (1995) 185–194; Statistics and Computing 6 (1996) 147–157); it is routinely available and can deliver substantial additional variance reduction. Finite-sample performance is illustrated in a sequential testing example. Connections are made with methods from the survey-sampling literature.

Contiguity and irreconcilable nonstandard asymptotics of statistical tests

Sun, 21 Aug 2011 20:37 EDT

Pranab K. Sen, Antonio C. Pedroso-de-Lima

Source: Braz. J. Probab. Stat., Volume 25, Number 3, 444--470.

Abstract:
Wald-type test statistics based on asymptotically normally distributed estimators (not necessarily maximum likelihood estimation or best asymptotically normal) provides an easy access to have tests for statistical hypotheses, far beyond the parametric paradigms. The methodological perspectives rest on a basic consistent asymptotic normal (CAN) condition which is interrelated to the well-known local asymptotic normality (LAN) condition. Contiguity of probability measures facilitates the C(L)AN condition in a relatively easier way. For many regular families of distributions, when statistical hypotheses do not involve nonstandard constraints, verification of contiguity of probability measures is facilitated by the well-known LeCam’s First Lemma [see Hájek, Šidák and Sen Theory of Rank Tests (1999), Chapter 7]. For nonregular families, though contiguity may hold under different setups, CAN estimators are not fully exploitable in the Wald type testing theory. This simple feature is illustrated by a two-parameter exponential model. Guided by this simple example, mixture of distributions are appraised in the context of Wald-type tests and the so-called χ̅ 2 - and E̅ -test theory is thoroughly appraised. A general result on counter examples is presented in detail.

An expansion for self-interacting random walks

Fri, 11 Nov 2011 15:26 EST

Remco van der Hofstad, Mark Holmes

Source: Braz. J. Probab. Stat., Volume 26, Number 1, 1--55.

Abstract:
We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions.

On data-based selection of summary measures from repeated measurements

Fri, 11 Nov 2011 15:26 EST

Ib M. Skovgaard, Torben Martinussen

Source: Braz. J. Probab. Stat., Volume 26, Number 1, 56--70.

Abstract:
Univariate analysis of variance of a good summary measure, or two, may provide a simple and effective way of analyzing repeated measurements. It is shown here that selection of a linear summary measure on the basis of inspection of the total sample of response curves, leads to valid F -tests in the subsequent analysis of variance. The selection may also be based on residuals from a base model, rather than on the raw data. The treatments should, however, be blinded in this summary measure selection step, that is, the inspection of the sample of curves (or residuals) and the selection of the summary measure may not rely on which responses stem from which treatment groups. It is advocated as a convenient and often effective method to use the first principal component from the total sample of curves as the first summary measure. The main mathematical result of the paper is a simple proof of the validity of the F -tests for linear summary measures selected in this way, provided data are multivariate normally distributed. Alternatively, permutation tests may be used to provide a distribution free reference distribution for the F -statistic. Two examples illustrate the method.

The multiplicative heteroscedastic Von Bertalanffy model

Fri, 11 Nov 2011 15:26 EST

Carlos Alberto Ribeiro Diniz, Francisco Louzada-Neto, Lia Hanna Martins Morita

Source: Braz. J. Probab. Stat., Volume 26, Number 1, 71--81.

Abstract:
In this work, we propose a heteroscedastic Von Bertalanffy growth model considering a multiplicative heteroscedastic dispersion matrix. All estimates were obtained using a sampling based approach, which allows information to be input beforehand with lower computational effort. Simulations were carried out in order to verify some frequentist properties of the estimation procedure in the presence of small and moderate sample sizes. The methodology is illustrated on a real Kubbard female chicken corporeal weight dataset.

A note on the Berman condition

Fri, 11 Nov 2011 15:26 EST

Rolf Turner, Patrick Chareka

Source: Braz. J. Probab. Stat., Volume 26, Number 1, 82--87.

Abstract:
It is established that if a time series satisfies the Berman condition, and another related (summability) condition, the result of filtering that series through a certain type of filter also satisfies the two conditions. In particular it follows that if X t satisfies the two conditions and if X t and a t are related by an invertible ARMA model, then the a t satisfy the two conditions.

The beta power distribution

Fri, 11 Nov 2011 15:26 EST

Gauss Moutinho Cordeiro, Rejane dos Santos Brito

Source: Braz. J. Probab. Stat., Volume 26, Number 1, 88--112.

Abstract:
The power distribution is defined as the inverse of the Pareto distribution. We study in full detail a distribution so-called the beta power distribution. We obtain analytical forms for its probability density and hazard rate functions. Explicit expressions are derived for the moments, probability weighted moments, moment generating function, mean deviations, Bonferroni and Lorenz curves, moments of order statistics, entropy and reliability. We estimate the parameters by maximum likelihood. The practicability of the model is illustrated in two applications to real data.

A simple formula for asymptotic distributional risk of some estimators

Mon, 23 Jan 2012 09:15 EST

Sévérien Nkurunziza

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 113--122.

Abstract:
In this article, we are interested in deriving the asymptotic distributional risk function of a class of estimator concerning the mean parameter matrix of matrices variate random sample. The proposed result is useful in decision theory, more precisely in risk analysis of a class of some robust estimators such as Stein-rule types estimators.

Test, estimation and model comparison for the meiosis I nondisjunction fraction in trisomies

Mon, 23 Jan 2012 09:15 EST

Vanessa L. Silva, Rosangela H. Loschi

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 123--148.

Abstract:
Trisomies are numerical chromosomal anomalies (aneuploidies) which are common causes of mental retardation, pregnancy losses and fetal death. The determination of the meiosis I nondisjunction fraction plays an important role in the identification of possible factors which could generate such aneuploidies. In this article, more flexible misclassification models for the number of peaks in a polymorphic locus of trisomic individuals are considered. They are compared to some others proposed in the literature. Estimation and tests for the nondisjunction fraction in meiosis I and for the misclassification errors are introduced extending previous works. Using the Decision Theory approach, we also build a criterion for making decisions under Jeffreys and Pereira–Stern tests. We apply the results to Down Syndrome data that is the most prevalent trisomy in humans.

Cornish–Fisher expansions for sample autocovariances and other functions of sample moments of linear processes

Mon, 23 Jan 2012 09:15 EST

Christopher S. Withers, Saralees Nadarajah

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 149--166.

Abstract:
We give Cornish–Fisher expansions for general smooth functions of the sample cross-moments of a stationary linear process. Examples include the distributions of the sample mean, the sample autocovariance and the sample autocorrelation.

GMM versus GQL inferences in semiparametric linear dynamic mixed models

Mon, 23 Jan 2012 09:15 EST

R. Prabhakar Rao, Brajendra Sutradhar, V. N. Pandit

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 167--177.

Abstract:
Linear dynamic mixed models are commonly used for continuous panel data analysis in economic statistics. There exists generalized method of moments (GMM) and generalized quasi-likelihood (GQL) inferences for binary and count panel data models, the GQL estimation approach being more efficient than the GMM approach. The GMM and GQL estimating equations for the linear dynamic mixed model can not, however, be obtained from the respective estimating equations under the nonlinear models for binary and count data. In this paper, we develop the GMM and GQL estimation approaches for the linear dynamic mixed models and demonstrate that the GQL approach is more efficient than the GMM approach, also under such linear models. This makes the GQL approach uniformly more efficient than the GMM approach in estimating the parameters of both linear and nonlinear dynamic mixed models.

The gamma beta ratio distribution

Mon, 23 Jan 2012 09:15 EST

Saralees Nadarajah

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 178--207.

Abstract:
The important problem of the ratio of gamma and beta distributed random variables is considered. Six motivating applications (from efficiency modeling, income modeling, clinical trials, hydrology, reliability and modeling of infectious diseases) are discussed. Exact expressions are derived for the probability density function, cumulative distribution function, hazard rate function, shape characteristics, moments, factorial moments, variance, skewness, kurtosis, conditional moments, L moments, characteristic function, mean deviation about the mean, mean deviation about the median, Bonferroni curve, Lorenz curve, percentiles, order statistics and the asymptotic distribution of the extreme values. Estimation procedures by the methods of moments and maximum likelihood are provided and their performances compared by simulation. For maximum likelihood estimation, the Fisher information matrix is derived and the case of censoring is considered. Finally, an application is discussed for efficiency of warning-time systems.

Properties of convergence of a fuzzy set estimator of the density function

Mon, 23 Jan 2012 09:15 EST

Jesús A. Fajardo, Ricardo R. Ríos, Luis A. Rodríguez

Source: Braz. J. Probab. Stat., Volume 26, Number 2, 208--217.

Abstract:
In this paper we establish the almost sure, in law, and uniform convergence over compact subsets on ℝ of a fuzzy set estimator of the density function, based on n i.i.d. random variable.

Group selection in high-dimensional partially linear additive models

Thu, 05 Apr 2012 09:23 EDT

Fengrong Wei

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 219--243.

Abstract:
We consider the problem of simultaneous variable selection and estimation in partially linear additive models with a large number of grouped variables in the linear part and a large number of nonparametric components. In our problem, the number of grouped variables may be larger than the sample size, but the number of important groups is “small” relative to the sample size. We apply the adaptive group Lasso to select the important groups, using spline bases to approximate the nonparametric components and the group Lasso to obtain an initial consistent estimator. Under appropriate conditions, it is shown that, the group Lasso selects the number of groups which is comparable with the underlying important groups and is estimation consistent, the adaptive group Lasso selects the correct important groups with probability converging to one as the sample size increases and is selection consistent. The results of simulation studies show that the adaptive group Lasso procedure works well with samples of moderate size. A real example is used to illustrate the application of the proposed penalized method.

A note on the robustness of a full Bayesian method for nonignorable missing data analysis

Thu, 05 Apr 2012 09:23 EDT

Zhiyong Zhang, Lijuan Wang

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 244--264.

Abstract:
A full Bayesian method utilizing data augmentation and Gibbs sampling algorithms is presented for analyzing nonignorable missing data. The discussion focuses on a simplified selection model for regression analysis. Regardless of missing mechanisms, it is assumed that missingness only depends on the missing variable itself. Simulation results demonstrate that the simplified selection model can recover regression model parameters under both correctly specified situations and many misspecified situations. The method is also applied to analyzing a training intervention data set with missing data.

Some Poisson mixtures distributions with a hyperscale parameter

Thu, 05 Apr 2012 09:23 EDT

Stéphane Laurent

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 265--278.

Abstract:
We mainly investigate certain mixtures of Poisson distributions with a scale parameter in the mixing distribution. They help us to derive the bivariate Poisson mixtures arising from the prior and posterior predictive distributions in the semi-conjugate family defined by Laurent and Legrand ( ESAIM Probab. Stat. (2011) DOI:10.1051/ps/2010018) for the “two Poisson samples” model, which contains in particular the reference prior when the parameter of interest is the ratio of the two Poisson rates.

A note on Bayesian robustness for count data

Thu, 05 Apr 2012 09:23 EDT

Jairo A. Fúquene, Moises Delgado

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 279--287.

Abstract:
The usual Bayesian approach for count data is Gamma/Poisson conjugate analysis. However, in this conjugate analysis the influence of the prior distribution could be dominant even when prior and likelihood are in conflict. Our proposal is an analysis based on the Cauchy prior for natural parameter in exponential families. In this work, we show that the Cauchy/Poisson posterior model is a robust model for count data in contrast with the usual conjugate Bayesian approach Gamma/Poisson model. We use the polynomial tails comparison theorem given in ( Bayesian Anal. 4 (2009) 817–843) that gives easy-to-check conditions to ensure prior robustness. In short, this means that when the location of the prior and the bulk of the mass of the likelihood get further apart (a situation of conflict between prior and likelihood information), Bayes theorem will cause the posterior distribution to discount the prior information. Finally, we analyze artificial data sets to investigate the robustness of the Cauchy/Poisson model.

On the copula for multivariate extreme value distributions

Thu, 05 Apr 2012 09:23 EDT

Marco Aurélio Sanfins, Glauco Valle

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 288--305.

Abstract:
We show that all multivariate extreme value distributions, which are the possible weak limits of the K largest order statistics of i.i.d. samples, have the same copula, the so called K -extremal copula. This copula and its density are described through exact expressions. We also study measures of dependence, we obtain a weak convergence result and we propose a simulation algorithm for the K -extremal copula.

Identifiability of zero-inflated Poisson models

Thu, 05 Apr 2012 09:23 EDT

Chin-Shang Li

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 306--312.

Abstract:
Zero-inflated Poisson (ZIP) models, which are mixture models, have been popularly used for count data that often contain large numbers of zeros, but their identifiability has not yet been thoroughly explored. In this work, we systematically investigate the identifiability of the ZIP models under a number of different assumptions. More specifically, we show the identifiability of a parametric ZIP model in which the incidence probability p ( x ) and Poisson mean λ ( x ) are modeled parametrically as p ( x ) = exp( β 0 + β 1 x )/[1 + exp( β 0 + β 1 x )] and λ ( x ) = exp( α 0 + α 1 x ) for x being a continuous covariate in a closed interval. A semiparametric ZIP regression model is shown to be identifiable in which (i) p ( x ) = exp( β 0 + β 1 x )/[1 + exp( β 0 + β 1 x )] and λ ( x ) = exp[ s ( x )], (ii) p ( x ) = exp[ r ( x )]/{1 + exp[ r ( x )]} and λ ( x ) = exp( α 0 + α 1 x ), or (iii) p ( x ) = exp[ r ( x )]/{1 + exp[ r ( x )]} and λ ( x ) = exp[ s ( x )] for r ( x ) and s ( x ) being unspecified smooth functions.

The polysurvival model with long-term survivors

Thu, 05 Apr 2012 09:23 EDT

Josmar Mazucheli, Francisco Louzada, Jorge A. Achcar

Source: Braz. J. Probab. Stat., Volume 26, Number 3, 313--324.

Abstract:
Long-term survival models have historically been considered for analyzing time-to-event data with long-term survivors fraction. However, situations in which a fraction (1 − p ) of systems is subject to failure from independent competing causes of failure, while the remaining proportion p is cured or has not presented the event of interest during the time period of the study, have not been fully considered in the literature. In order to accommodate such situations, we present in this paper a new long-term survival model. Maximum likelihood estimation procedure is discussed as well as interval estimation and hypothesis tests. A real dataset illustrates the methodology.

Preface

Tue, 03 Jul 2012 08:57 EDT

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 325--326.

Bayesian analysis based on the Jeffreys prior for the hyperbolic distribution

Tue, 03 Jul 2012 08:57 EDT

Thaís C. O. Fonseca, Helio S. Migon, Marco A. R. Ferreira

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 327--343.

Abstract:
In this work, we develop Bayesian analysis based on the Jeffreys prior for the hyperbolic family of distributions. It is usually difficult to estimate the four parameters in this class: to be reliable the maximum likelihood estimator typically requires large sample sizes of the order of thousands of observations. Moreover, improper prior distributions may lead to improper posterior distributions, whereas proper prior distributions may dominate the analysis. Here, we show through a simulation study that Bayesian methods based on Jeffreys prior provide reliable point and interval estimators. Moreover, this simulation study shows that for the absolute loss function Bayesian estimators compare favorably to maximum likelihood estimators. Finally, we illustrate with an application to real data that our methodology allows for parameter estimation with remarkable good properties even for a small sample size.

Latent residual analysis in binary regression with skewed link

Tue, 03 Jul 2012 08:57 EDT

Rafael B. A. Farias, Marcia D. Branco

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 344--357.

Abstract:
Model diagnostics is an integral part of model determination and an important part of the model diagnostics is residual analysis. We adapt and implement residuals considered in the literature for the probit, logistic and skew-probit links under binary regression. New latent residuals for the skew-probit link are proposed here. We have detected the presence of outliers using the residuals proposed here for different models in a simulated dataset and a real medical dataset.

Bayesian statistics with a smile: A resampling–sampling perspective

Tue, 03 Jul 2012 08:57 EDT

Hedibert F. Lopes, Nicholas G. Polson, Carlos M. Carvalho

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 358--371.

Abstract:
In this paper we develop a simulation-based approach to sequential inference in Bayesian statistics. Our resampling–sampling perspective provides draws from posterior distributions of interest by exploiting the sequential nature of Bayes theorem. Predictive inferences are a direct byproduct of our analysis as are marginal likelihoods for model assessment. We illustrate our approach in a hierarchical normal-means model and in a sequential version of Bayesian lasso. This approach provides a simple yet powerful framework for the construction of alternative posterior sampling strategies for a variety of commonly used models.

Bayesian heavy-tailed models and conflict resolution: A review

Tue, 03 Jul 2012 08:57 EDT

Anthony O’Hagan, Luis Pericchi

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 372--401.

Abstract:
We review a substantial literature, spanning 50 years, concerning the resolution of conflicts using Bayesian heavy-tailed models. Conflicts arise when different sources of information about the model parameters (e.g., prior information, or the information in individual observations) suggest quite different plausible regions for those parameters. Traditional Bayesian models based on normal distributions or other conjugate structures typically resolve conflicts by centring the posterior at some compromise position, but this is not a realistic resolution when it means that the posterior is then in conflict with the different information sources. Bayesian modelling with heavy-tailed distributions has been shown to produce more reasonable conflict resolution, typically by favouring one source of information over the other. The less favoured source is ultimately wholly or partially rejected as the conflict becomes increasingly extreme. The literature reviewed here provides formal proofs of conflict resolution by asymptotic rejection of some information sources. Results are given for a variety of models, from the simplest case of a single observation relating to a single location parameter up to models with many location parameters, location and scale parameters, or other kinds of parameters. However, these results do not begin to address models of the kind of complexity that are routinely used in practical Bayesian modelling. In addition to reviewing the available theory, we also identify clearly the gaps in the literature that need to be filled in order for modellers to be able to develop applications with appropriate “built-in robustness.”

Stochastic volatility in mean models with heavy-tailed distributions

Tue, 03 Jul 2012 08:57 EDT

Carlos A. Abanto-Valle, Helio S. Migon, Victor H. Lachos

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 402--422.

Abstract:
A stochastic volatility in mean (SVM) model using the class of symmetric scale mixtures of normal (SMN) distributions is introduced in this article. The SMN distributions form a class of symmetric thick-tailed distributions that includes the normal one as a special case, providing a robust alternative to estimation in SVM models in the absence of normality. A Bayesian method via Markov-chain Monte Carlo (MCMC) techniques is used to estimate parameters. The deviance information criterion (DIC) and the Bayesian predictive information criteria (BPIC) are calculated to compare the fit of distributions. The method is illustrated by analyzing daily stock return data from the São Paulo Stock, Mercantile & Futures Exchange index (IBOVESPA). According to both model selection criteria as well as out-of-sample forecasting, we found that the SVM model with slash distribution provides a significant improvement in model fit as well as prediction for the IBOVESPA data over the usual normal model.

Predictive construction of priors in Bayesian nonparametrics

Tue, 03 Jul 2012 08:57 EDT

Sandra Fortini, Sonia Petrone

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 423--449.

Abstract:
The characterization of models and priors through a predictive approach is a fundamental problem in Bayesian statistics. In the last decades, it has received renewed interest, as the basis of important developments in Bayesian nonparametrics and in machine learning. In this paper, we review classical and recent work based on the predictive approach in these areas. Our focus is on the predictive construction of priors for Bayesian nonparametric inference, for exchangeable and partially exchangeable sequences. Some results are revisited to shed light on theoretical connections among them.

Test procedures based on combination of Bayesian evidences for $H_{0}$

Tue, 03 Jul 2012 08:57 EDT

Rosangela H. Loschi, Cristiano C. Santos, Reinaldo B. Arellano-Valle

Source: Braz. J. Probab. Stat., Volume 26, Number 4, 450--473.

Abstract:
We introduce two procedures for testing which are based on pooling the posterior evidence for the null hypothesis provided by the full Bayesian significance test and the posterior probability for the null hypothesis. Although the proposed procedures can be used in more general situations, we focus attention in tests for a precise null hypothesis. We prove that the proposed procedure based on the linear operator is a Bayes rule. We also verify that it does not lead to the Jeffreys–Lindley paradox. For a precise null hypothesis, we prove that the procedure based on the logarithmic operator is a generalization of Jeffreys test. We apply the results to some well-known probability families. The empirical results show that the proposed procedures present good performances. As a by-product we obtain tests for normality under the skew-normal one.

Beta generalized distributions and related exponentiated models: A Bayesian approach

Tue, 16 Oct 2012 09:37 EDT

Jorge A. Achcar, Emílio A. Coelho-Barros, Gauss M. Cordeiro

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 1--19.

Abstract:
We introduce a Bayesian analysis for beta generalized distributions and related exponentiated models. We review the exponentiated exponential, exponentiated Weibull and beta generalized exponential distributions. These distributions have been proposed as alternative extensions of the gamma and Weibull distributions in the analysis of lifetime data. Some posterior summaries of interest are obtained using Monte Carlo Markov chain (MCMC) methods. An application to a real data set is given to illustrate the potentiality of the Bayesian analysis.

Precise asymptotics for products of sums and U-statistics

Tue, 16 Oct 2012 09:37 EDT

Zhongquan Tan

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 20--30.

Abstract:
Let $\{X,X_{i},i\geq 1\}$ be a sequence of independent and identically distributed positive random variables with $E(X)=\mu >0$, $\operatorname{Var}(X)<\infty$. Put $S_{n}=\sum_{i=1}^{n}X_{i}$ and let $g(x)$ be a positive and differentiable function defined on $(0,+\infty)$ satisfying some mild conditions. We prove that, for any $s>1$, \[\lim_{\varepsilon\rightarrow0}\varepsilon^{1/s}\sum_{n=1}^{\infty}g'(n)P\Biggl\{\Biggl|\log\Biggl(\prod_{j=1}^{n}\frac{S_{j}}{j\mu}\Biggr)\Biggr|\geq\varepsilon\sqrt{n}g^{s}(n)\Biggr\}=E|N|^{1/s},\] where $N$ is a standard normal random variable. This result was also extended to product of U-statistics.

The exponentiated Kumaraswamy distribution and its log-transform

Tue, 16 Oct 2012 09:37 EDT

Artur J. Lemonte, Wagner Barreto-Souza, Gauss M. Cordeiro

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 31--53.

Abstract:
The paper by Kumaraswamy ( Journal of Hydrology 46 (1980) 79–88) introduced a probability distribution for double bounded random processes which has considerable attention in hydrology and related areas. Based on this distribution, we propose a generalization of the Kumaraswamy distribution refereed to as the exponentiated Kumaraswamy distribution. We derive the moments, moment generating function, mean deviations, Bonferroni and Lorentz curves, density of the order statistics and their moments. We also present a related distribution, so-called the log-exponentiated Kumaraswamy distribution, which extends the generalized exponential ( Aust. N. Z. J. Stat. 41 (1999) 173–188) and double generalized exponential ( J. Stat. Comput. Simul. 80 (2010) 159–172) distributions. We discuss maximum likelihood estimation of the model parameters. In applications to real data sets, we show that the log-exponentiated Kumaraswamy model can be used quite effectively in analyzing lifetime data.

On some fundamental aspects of polyominoes on random Voronoi tilings

Tue, 16 Oct 2012 09:37 EDT

Leandro P. R. Pimentel

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 54--69.

Abstract:
Consider a Voronoi tiling of $\mathbb{R} ^{d}$ based on a realization of an inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected union of Voronoi tiles. In this paper we provide tail bounds for the number of boxes that are intersected by a Voronoi polyomino, and vice-versa. These results will be crucial to analyze self-avoiding paths, greedy polyominoes and first-passage percolation models on Voronoi tilings and on the dual graph, named the Delaunay triangulation [Asymptotics for first-passage times on Delaunay triangulations (2011) Preprint, Greedy Polyominoes and first-passage times on random Voronoi tilings (2012) Preprint].

A fully Bayesian parametric approach for cytogenetic dosimetry

Tue, 16 Oct 2012 09:37 EDT

Carlos Daniel Paulino, Giovani L. Silva, Márcia Branco

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 70--83.

Abstract:
This paper describes a new statistical analysis strategy to problems of cytogenetic dosimetry involving ordinal polythomous responses. Models relating the multivariate response to dose take the data ordinality into account and are analysed in a fully Bayesian fashion in the application here considered. In particular, these models are compared in order to select the best one for purposes of drawing inferences of interest and dose prediction is naturally addressed by its practical importance. This work was motivated by an in vitro experimental study on radiation exposure of human blood cell cultures, previously analysed in the literature by other methods, but its interest holds in many other applications of the biological and environmental field involving data sets yielded from the same type of assays for genetic damage.

The exp-$G$ family of probability distributions

Tue, 16 Oct 2012 09:37 EDT

Wagner Barreto-Souza, Alexandre B. Simas

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 84--109.

Abstract:
In this paper we introduce a new method to add a parameter to a family of distributions. The additional parameter is completely studied and a full description of its behaviour in the distribution is given. We obtain several mathematical properties of the new class of distributions such as Kullback–Leibler divergence, Shannon entropy, moments, order statistics, estimation of the parameters and inference for large sample. Further, we show that the new distribution has the reference distribution as special case, and that the usual inference procedures also hold in this case. We present a comprehensive study of two special cases of the exp-$G$ class: exp-Weibull and exp-beta distributions. Further, an application to the real data set is presented. This family also opens a wide variety of research, as the authors may develop its special cases in full detail.

A note on the parameterization of multivariate skewed-normal distributions

Tue, 16 Oct 2012 09:37 EDT

Luis M. Castro, Ernesto San Martín, Reinaldo B. Arellano-Valle

Source: Braz. J. Probab. Stat., Volume 27, Number 1, 110--115.

Abstract:
Azzalini’s skew-normal distribution is obtained through a conditional reduction of a multivariate normal distribution parameterized with a correlation matrix. It seems natural that when the parameterization of that multivariate normal distribution is complexified, more flexible skew-normal distributions could be obtained. In this note this specification strategy, previously explored by Azzalini [ Scand. J. Stat. 33 (2006) 561–574] among many other authors, is formally analyzed through an identification analysis.

Acceptance sampling plans from truncated life tests based on the Marshall–Olkin extended exponential distribution for percentiles

Thu, 21 Feb 2013 08:58 EST

G. Srinivasa Rao

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 117--132.

Abstract:
In this article, acceptance sampling plans are developed for the Marshall–Olkin extended exponential distribution percentiles when the life test is truncated at a pre-specified time. The minimum sample size necessary to ensure the specified life percentile is obtained under a given customer’s risk. The operating characteristic values (and curves) of the sampling plans as well as the producer’s risk are presented. Two examples with real data sets are also given as illustration.

A new extension of the Birnbaum–Saunders distribution

Thu, 21 Feb 2013 08:58 EST

Artur J. Lemonte

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 133--149.

Abstract:
In this paper, a new extension for the Birnbaum–Saunders distribution, which has been applied to the modeling of fatigue failure times and reliability studies, is introduced. The proposed model, called the Marshall–Olkin extended Birnbaum–Saunders distribution, arises based on the scheme introduced by Marshall and Olkin [ Biometrika 84 (1997) 641–652]. The maximum likelihood estimators and statistical inference for the new distribution parameters and influence diagnostic for the new distribution are presented. Finally, the proposed new distribution is applied to model three real data sets.

On a saddlepoint approximation to the Markov binomial distribution

Thu, 21 Feb 2013 08:58 EST

Jens Ledet Jensen

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 150--161.

Abstract:
A nonstandard saddlepoint approximation to the distribution of a sum of Markov dependent trials is introduced. The relative error of the approximation is studied, not only for the number of summands tending to infinity, but also for the parameter approaching the boundary of its definition range. A comparison is made with another recent study of Markov dependent trials.

Marshall–Olkin Esscher transformed Laplace distribution and processes

Thu, 21 Feb 2013 08:58 EST

Dais George, Sebastian George

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 162--184.

Abstract:
In this article we consider a class of asymmetric distributions which belongs to one parameter regular exponential family. The Marshall–Olkin version of this family is also considered. Various properties are examined. Applications of these models in time series analysis are discussed. We also consider an application of Marshall–Olkin Esscher transformed Laplace distribution in financial modeling. A comparative study shows that Marshall–Olkin Esscher transformed Laplace distribution is a better fit to our data compared to asymmetric Laplace and Esscher transformed Laplace distributions.

The beta generalized logistic distribution

Thu, 21 Feb 2013 08:58 EST

Alice L. Morais, Gauss M. Cordeiro, Audrey H. M. A. Cysneiros

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 185--200.

Abstract:
For the first time, a four-parameter beta generalized logistic distribution is obtained by compounding the beta and generalized logistic distributions. The new model extends some well-known distributions and its shape is quite flexible, specially the skewness and the tail weights, due to the extra shape parameters. We obtain general expansions for the moment generating and quantile functions. The estimation of the parameters is investigated by maximum likelihood. An application to a real data set is given to show the flexibility and potentiality of our distribution.

On a link between a species survival time in an evolution model and the Bessel distributions

Thu, 21 Feb 2013 08:58 EST

Hervé Guiol, Fábio P. Machado, Rinaldo Schinazi

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 201--209.

Abstract:
We consider a stochastic model for species evolution. A new species is born at rate $\lambda$ and a species dies at rate $\mu$. A random number, sampled from a given distribution $F$, is associated with each new species and assumed as its fitness, at the time of birth. Every time there is a death event, the species that is killed is the one with the smallest fitness. We consider the (random) survival time of a species with a given fitness $f$. We show that the survival time distribution depends crucially on whether $f<f_{c}$, $f=f_{c}$ or $f>f_{c}$ where $f_{c}$ is a critical fitness that is computed explicitly.

Score-type statistics in pattern classification

Thu, 21 Feb 2013 08:58 EST

Manoel R. Sena Jr., Abraão D. C. Nascimento, Gauss M. Cordeiro, Lúcia P. Barroso

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 210--226.

Abstract:
Statistical classification methods based on score statistics have received a considerable attention in recent years. The use of these methodologies requires that asymptotic properties relative to such measures are satisfied. In this context, the classification error rates generally present biased values to the nominal level when submitted to small or moderate sample sizes. However, Nelson, Turin and Hastie [ Journal of Pattern Recognition and Artificial Intelligence 8 (1994) 749–770] proposed a successful classification method based on score statistic described asymptotically by a chi-square law. That proposal presented good results for several sample sizes. On the other hand, stochastic measures with exact distributions described by beta and Hotelling’s $\mathcal{T}^{2}$ laws have also been employed in such situations. This paper presents two Bartlett-type corrections for score statistics considering the method proposed by Cordeiro and Ferrari [ J. Statist. Plann. Inference 71 (1998) 261–269]. Moreover, Monte Carlo experiments are performed in order to compare the corrected statistics to their respective noncorrected versions and to a classic classifier defined on the nonmodified score statistic. In a confirmatory sense, the proposed methodology is applied to actual signature data, obtained by the Electrical Engineering and Computer Department from the State University of Campinas (UNICAMP, Brazil).

Finite exclusion process and independent random walks

Thu, 21 Feb 2013 08:58 EST

E. D. Andjel

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 227--244.

Abstract:
We show that the total variational distance between a process of two particles interacting by exclusion and a process of two independent particles goes to $0$ as time goes to infinity, when the underlying one particle system is a symmetric random walk on $\mathbb{Z}^{d}$ with finite second moments. Upper bounds for the speed of convergence are given.

CADEM: A conditional augmented data EM algorithm for fitting one parameter probit models

Thu, 21 Feb 2013 08:58 EST

C. L. N. Azevedo, D. F. Andrade

Source: Braz. J. Probab. Stat., Volume 27, Number 2, 245--262.

Abstract:
In this article we develop an estimation method based on the augmented data scheme and EM/SEM (Stochastic EM) algorithms for fitting one-parameter probit (Rasch) IRT (Item Response Theory) models. Instead of using the S steps of the SEM algorithm, that is, instead of simulating values for the unobserved variables (augmented data and the latent traits), we consider the conditional expectations of a set of unobserved variables on the other set of unobserved variables, the current estimates of the parameters and the observed data, based on the full conditional distributions from the Gibbs sampling algorithm. Our method, named the CADEM algorithm (conditional augmented data EM), presents straightforward E steps, which avoid the need to evaluate the usual integrals, also facilitating the M steps, without the need to use numerical methods of optimization. We use the CADEM algorithm to obtain both maximum likelihood estimates and maximum a posteriori estimates of the difficulty parameters for the one-parameter probit (Rasch) model. Also, we obtain estimates for the latent traits, based on conditional expectations. In addition, we show how to calculate the associated standard errors. Some directions are provided to extend our approach to other IRT models. In this respect, we perform a simulation study to compare the estimation methods. The results indicated that our approach is quite comparable to the usual marginal maximum likelihood (MML) and Gibbs sampling methods (GS) in terms of parameter recovery. However, CADEM is as fast as MML and as flexible as GS.

Preface

Tue, 28 May 2013 09:08 EDT

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 263--264.

Predicting dependent binary outcomes through logistic regressions and meta-elliptical copulas

Tue, 28 May 2013 09:08 EDT

Christian Genest, Aristidis K. Nikoloulopoulos, Louis-Paul Rivest, Mathieu Fortin

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 265--284.

Abstract:
The authors consider copula models for vectors of binary response variables having marginal distributions that depend on covariates through logistic regressions. They show how to test for residual pairwise dependence between responses, given the explanatory variables. The procedure they propose is based on the score statistic derived from the assumed copula structure under the alternative. The authors further argue that conditional dependence can be conveniently modelled with meta-elliptical copulas, which offer a wide range of positive and negative degrees of association. They call on a composite likelihood to estimate the copula parameters and they provide standard error estimates of the same via linearization. They illustrate their results with Canadian data on the presence or absence of various log grades in trees.

Construction of multivariate dispersion models

Tue, 28 May 2013 09:08 EDT

Bent Jørgensen

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 285--309.

Abstract:
We consider methods for constructing multivariate dispersion models, illustrated by examples. Such methods are motivated by the need for good regression modelling of multivariate nonnormal correlated data, which requires multivariate distributions with a flexible correlation structure. We first review existing methods for constructing multivariate proper dispersion models, involving quadratic forms of deviance residuals in the style of the multivariate normal density, which we illustrate by a multivariate hyperbola distribution. We develop an extended convolution method for constructing multivariate exponential dispersion models, designed to create a fully flexible covariance structure, which we illustrate by two bivariate gamma distributions. We develop a similar technique for constructing multivariate extreme dispersion models for extremes and survival data, and introduce new bivariate logistic and Gumbel distributions.

Extendibility of Marshall–Olkin distributions and inverse Pascal triangles

Tue, 28 May 2013 09:08 EDT

Jan-Frederik Mai, Matthias Scherer

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 310--321.

Abstract:
Necessary and sufficient conditions are derived on the parameters of a $d$-dimensional random vector with Marshall–Olkin distribution to be extendible to an infinite exchangeable sequence. Interpreted differently, this result allows to decide if the respective multivariate exponential distribution can be constructed by means of a model with conditionally independent and identically distributed components. The proof makes use of the solution of the truncated Hausdorff moment problem and a reparameterization of the Marshall–Olkin distribution.

Copulas related to Manneville–Pomeau processes

Tue, 28 May 2013 09:08 EDT

Sílvia R. C. Lopes, Guilherme Pumi

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 322--356.

Abstract:
In this work, we derive the copulas related to Manneville–Pomeau processes. We examine both bidimensional and multidimensional cases and derive some properties for the related copulas. Computational issues, approximations and random variate generation problems are addressed and simple numerical experiments to test the approximations developed are also performed. In particular, we propose an approximation to the derived copulas which we show to converge uniformly to the true one. To illustrate the usefulness of the theory, we derive a fast procedure to estimate the underlying parameter in Manneville–Pomeau processes.

Polyhazard models with dependent causes

Tue, 28 May 2013 09:08 EDT

Rodrigo Tsai, Luiz Koodi Hotta

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 357--376.

Abstract:
Polyhazard models constitute a flexible family for fitting lifetime data. The main advantages over single hazard models include the ability to represent hazard rate functions with unusual shapes and the ease of including covariates. The primary goal of this paper was to include dependence among the latent causes of failure by modeling dependence using copula functions. The choice of the copula function as well as the latent hazard functions results in a flexible class of survival functions that is able to represent hazard rate functions with unusual shapes, such as bathtub or multimodal curves, while also modeling local effects associated with competing risks. The model is applied to two sets of simulated data as well as to data representing the unemployment duration of a sample of socially insured German workers. Model identification and estimation are also discussed.

Manifold matching: Joint optimization of fidelity and commensurability

Tue, 28 May 2013 09:08 EDT

Carey E. Priebe, David J. Marchette, Zhiliang Ma, Sancar Adali

Source: Braz. J. Probab. Stat., Volume 27, Number 3, 377--400.

Abstract:
Fusion and inference from multiple and massive disparate data sources—the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies—has many and varied aspects which are currently the target of intense research and development. One aspect of the overall challenge is manifold matching—identifying embeddings of multiple disparate data spaces into the same low-dimensional space where joint inference can be pursued. We investigate this manifold matching task from the perspective of jointly optimizing the fidelity of the embeddings and their commensurability with one another, with a specific statistical inference exploitation task in mind. Our results demonstrate when and why our joint optimization methodology is superior to either version of separate optimization. The methodology is illustrated with simulations and an application in document matching.