When the data is distributed across multiple servers, lowering the communication cost between the servers (or workers) while solving the distributed learning problem is an important problem and is the focus of this paper. In particular, we propose a fast, and communication-efficient decentralized framework to solve the distributed machine learning (DML) problem. The proposed algorithm, Group Alternating Direction Method of Multipliers (GADMM) is based on the Alternating Direction Method of Multipliers (ADMM) framework. The key novelty in GADMM is that it solves the problem in a decentralized topology where at most half of the workers are competing for the limited communication resources at any given time. Moreover, each worker exchanges the locally trained model only with two neighboring workers, thereby training a global model with a lower amount of communication overhead in each exchange. We prove that GADMM converges to the optimal solution for convex loss functions, and numerically show that it converges faster and more communication-efficient than the state-of-the-art communication-efficient algorithms such as the Lazily Aggregated Gradient (LAG) and dual averaging, in linear and logistic regression tasks on synthetic and real datasets. Furthermore, we propose Dynamic GADMM (D-GADMM), a variant of GADMM, and prove its convergence under the time-varying network topology of the workers.

We consider a setting where multiple players sequentially choose among a common set of actions (arms). Motivated by an application to cognitive radio networks, we assume that players incur a loss upon colliding, and that communication between players is not possible. Existing approaches assume that the system is stationary. Yet this assumption is often violated in practice, e.g., due to signal strength fluctuations. In this work, we design the first multi-player Bandit algorithm that provably works in arbitrarily changing environments, where the losses of the arms may even be chosen by an adversary. This resolves an open problem posed by Rosenski et al. (2016).

Daniela Bertacchi, Fabio Zucca.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 426--438.

Abstract:
Given a branching random walk on a set $X$, we study its extinction probability vectors $mathbf{q}(cdot,A)$. Their components are the probability that the process goes extinct in a fixed $Asubseteq X$, when starting from a vertex $xin X$. The set of extinction probability vectors (obtained letting $A$ vary among all subsets of $X$) is a subset of the set of the fixed points of the generating function of the branching random walk. In particular here we are interested in the cardinality of the set of extinction probability vectors. We prove results which allow to understand whether the probability of extinction in a set $A$ is different from the one of extinction in another set $B$. In many cases there are only two possible extinction probability vectors and so far, in more complicated examples, only a finite number of distinct extinction probability vectors had been explicitly found. Whether a branching random walk could have an infinite number of distinct extinction probability vectors was not known. We apply our results to construct examples of branching random walks with uncountably many distinct extinction probability vectors.

Nicola Kistler, Adrien Schertzer, Marius A. Schmidt.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 414--425.

Abstract:
The Poisson clumping heuristic has lead Aldous to conjecture the value of the oriented first passage percolation on the hypercube in the limit of large dimensions. Aldous’ conjecture has been rigorously confirmed by Fill and Pemantle ( Ann. Appl. Probab. 3 (1993) 593–629) by means of a variance reduction trick. We present here a streamlined and, we believe, more natural proof based on ideas emerged in the study of Derrida’s random energy models.

Benjamin Arras, Ehsan Azmoodeh, Guillaume Poly, Yvik Swan.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 394--413.

Abstract:
In this paper we propose a new, simple and explicit mechanism allowing to derive Stein operators for random variables whose characteristic function satisfies a simple ODE. We apply this to study random variables which can be represented as linear combinations of (not necessarily independent) gamma distributed random variables. The connection with Malliavin calculus for random variables in the second Wiener chaos is detailed. An application to McKay Type I random variables is also outlined.

Jun Zhang, Yujie Gai, Xia Cui, Gaorong Li.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 370--393.

Abstract:
This paper studies the measure of symmetry or asymmetry of a continuous variable under the multiplicative distortion measurement errors setting. The unobservable variable is distorted in a multiplicative fashion by an observed confounding variable. First, two direct plug-in estimation procedures are proposed, and the empirical likelihood based confidence intervals are constructed to measure the symmetry or asymmetry of the unobserved variable. Next, we propose four test statistics for testing whether the unobserved variable is symmetric or not. The asymptotic properties of the proposed estimators and test statistics are examined. We conduct Monte Carlo simulation experiments to examine the performance of the proposed estimators and test statistics. These methods are applied to analyze a real dataset for an illustration.

Raj Kamal Maurya, Yogesh Mani Tripathi.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 345--369.

Abstract:
We consider estimation of the multicomponent stress-strength reliability under progressive Type II censoring under the assumption that stress and strength variables follow Burr XII distributions with a common shape parameter. Maximum likelihood estimates of the reliability are obtained along with asymptotic intervals when common shape parameter may be known or unknown. Bayes estimates are also derived under the squared error loss function using different approximation methods. Further, we obtain exact Bayes and uniformly minimum variance unbiased estimates of the reliability for the case common shape parameter is known. The highest posterior density intervals are also obtained. We perform Monte Carlo simulations to compare the performance of proposed estimates and present a discussion based on this study. Finally, two real data sets are analyzed for illustration purposes.

Erlandson F. Saraiva, Adriano K. Suzuki, Luís A. Milan.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 323--344.

Abstract:
In this paper, we introduce a Bayesian approach for clustering data using a sparse finite mixture model (SFMM). The SFMM is a finite mixture model with a large number of components $k$ previously fixed where many components can be empty. In this model, the number of components $k$ can be interpreted as the maximum number of distinct mixture components. Then, we explore the use of a prior distribution for the weights of the mixture model that take into account the possibility that the number of clusters $k_{mathbf{c}}$ (e.g., nonempty components) can be random and smaller than the number of components $k$ of the finite mixture model. In order to determine clusters we develop a MCMC algorithm denominated Split-Merge allocation sampler. In this algorithm, the split-merge strategy is data-driven and was inserted within the algorithm in order to increase the mixing of the Markov chain in relation to the number of clusters. The performance of the method is verified using simulated datasets and three real datasets. The first real data set is the benchmark galaxy data, while second and third are the publicly available data set on Enzyme and Acidity, respectively.

Danilo Covaes Nogarotto, Caio Lucidius Naberezny Azevedo, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 304--322.

Abstract:
The interest on the analysis of the zero–one augmented beta regression (ZOABR) model has been increasing over the last few years. In this work, we developed a Bayesian inference for the ZOABR model, providing some contributions, namely: we explored the use of Jeffreys-rule and independence Jeffreys prior for some of the parameters, performing a sensitivity study of prior choice, comparing the Bayesian estimates with the maximum likelihood ones and measuring the accuracy of the estimates under several scenarios of interest. The results indicate, in a general way, that: the Bayesian approach, under the Jeffreys-rule prior, was as accurate as the ML one. Also, different from other approaches, we use the predictive distribution of the response to implement Bayesian residuals. To further illustrate the advantages of our approach, we conduct an analysis of a real psychometric data set including a Bayesian residual analysis, where it is shown that misleading inference can be obtained when the data is transformed. That is, when the zeros and ones are transformed to suitable values and the usual beta regression model is considered, instead of the ZOABR model. Finally, future developments are discussed.

Uttam Bandyopadhyay, Shirsendu Mukherjee, Atanu Biswas.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 291--303.

Abstract:
In adaptive crossover design, our goal is to allocate more patients to a promising treatment sequence. The present work contains a very simple three period crossover design for two competing treatments where the allocation in period 3 is done on the basis of the data obtained from the first two periods. Assuming normality of response variables we use a reliability functional for the choice between two treatments. We calculate the allocation proportions and their standard errors corresponding to the possible treatment combinations. We also derive some asymptotic results and provide solutions on related inferential problems. Moreover, the proposed procedure is compared with a possible competitor. Finally, we use a data set to illustrate the applicability of the proposed design.

Mohsen Maleki, Arezo Hajrajabi, Reinaldo B. Arellano-Valle.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 273--290.

Abstract:
In this paper, we study the finite mixtures of autoregressive processes assuming that the distribution of innovations (errors) belongs to the class of scale mixture of skew-normal (SMSN) distributions. The SMSN distributions allow a simultaneous modeling of the existence of outliers, heavy tails and asymmetries in the distribution of innovations. Therefore, a statistical methodology based on the SMSN family allows us to use a robust modeling on some non-linear time series with great flexibility, to accommodate skewness, heavy tails and heterogeneity simultaneously. The existence of convenient hierarchical representations of the SMSN distributions facilitates also the implementation of an ECME-type of algorithm to perform the likelihood inference in the considered model. Simulation studies and the application to a real data set are finally presented to illustrate the usefulness of the proposed model.

Zhengwei Liu, Qi Li, Fukang Zhu.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 251--272.

Abstract:
To predict time series of counts with small values and remarkable fluctuations, an available model is the $r$ states random environment process based on the negative binomial thinning operator and the geometric marginal. However, we argue that the aforementioned model may suffer from the following two drawbacks. First, under the condition of no prior information, the overdispersed property of the geometric distribution may cause the predictions fluctuate greatly. Second, because of the constraints on the model parameters, some estimated parameters are close to zero in real-data examples, which may not objectively reveal the correlation relationship. For the first drawback, an $r$ states random environment process based on the binomial thinning operator and the Poisson marginal is introduced. For the second drawback, we propose a generalized $r$ states random environment integer-valued autoregressive model based on the binomial thinning operator to model fluctuations of data. Yule–Walker and conditional maximum likelihood estimates are considered and their performances are assessed via simulation studies. Two real-data sets are conducted to illustrate the better performances of the proposed models compared with some existing models.

Victor Coscrato, Rafael Izbicki, Rafael B. Stern.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 230--250.

Abstract:
Despite its common practice, statistical hypothesis testing presents challenges in interpretation. For instance, in the standard frequentist framework there is no control of the type II error. As a result, the non-rejection of the null hypothesis $(H_{0})$ cannot reasonably be interpreted as its acceptance. We propose that this dilemma can be overcome by using agnostic hypothesis tests, since they can control the type I and II errors simultaneously. In order to make this idea operational, we show how to obtain agnostic hypothesis in typical models. For instance, we show how to build (unbiased) uniformly most powerful agnostic tests and how to obtain agnostic tests from standard p-values. Also, we present conditions such that the above tests can be made logically coherent. Finally, we present examples of consistent agnostic hypothesis tests.

Israel Martínez-Hernández, Marc G. Genton.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 204--229.

Abstract:
As high-dimensional and high-frequency data are being collected on a large scale, the development of new statistical models is being pushed forward. Functional data analysis provides the required statistical methods to deal with large-scale and complex data by assuming that data are continuous functions, for example, realizations of a continuous process (curves) or continuous random field (surfaces), and that each curve or surface is considered as a single observation. Here, we provide an overview of functional data analysis when data are complex and spatially correlated. We provide definitions and estimators of the first and second moments of the corresponding functional random variable. We present two main approaches: The first assumes that data are realizations of a functional random field, that is, each observation is a curve with a spatial component. We call them spatial functional data . The second approach assumes that data are continuous deterministic fields observed over time. In this case, one observation is a surface or manifold, and we call them surface time series . For these two approaches, we describe software available for the statistical analysis. We also present a data illustration, using a high-resolution wind speed simulated dataset, as an example of the two approaches. The functional data approach offers a new paradigm of data analysis, where the continuous processes or random fields are considered as a single entity. We consider this approach to be very valuable in the context of big data.

David A. C. Mollinedo.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 188--201.

Abstract:
We consider the stochastic transport equation with a possibly unbounded Hölder continuous vector field. Well-posedness is proved, namely, we show existence, uniqueness and strong stability of $W^{1,p}$-weak solutions.

Saralees Nadarajah, Yuancheng Si.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 183--187.

Abstract:
Da Paz, Balakrishnan and Bazan [Braz. J. Probab. Stat. 33 (2019), 455–479] introduced the L-logistic distribution, studied its properties including estimation issues and illustrated a data application. This note derives a closed form expression for moment properties of the distribution. Some computational issues are discussed.

Mohd Arshad, Omer Abdalghani.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 167--182.

Abstract:
Suppose that $pi_{1},pi_{2},ldots ,pi_{k}$ be $k(geq2)$ independent exponential populations having unknown location parameters $mu_{1},mu_{2},ldots,mu_{k}$ and known scale parameters $sigma_{1},ldots,sigma_{k}$. Let $mu_{[k]}=max {mu_{1},ldots,mu_{k}}$. For selecting the population associated with $mu_{[k]}$, a class of selection rules (proposed by Arshad and Misra [ Statistical Papers 57 (2016) 605–621]) is considered. We consider the problem of estimating the location parameter $mu_{S}$ of the selected population under the criterion of the LINEX loss function. We consider three natural estimators $delta_{N,1},delta_{N,2}$ and $delta_{N,3}$ of $mu_{S}$, based on the maximum likelihood estimators, uniformly minimum variance unbiased estimator (UMVUE) and minimum risk equivariant estimator (MREE) of $mu_{i}$’s, respectively. The uniformly minimum risk unbiased estimator (UMRUE) and the generalized Bayes estimator of $mu_{S}$ are derived. Under the LINEX loss function, a general result for improving a location-equivariant estimator of $mu_{S}$ is derived. Using this result, estimator better than the natural estimator $delta_{N,1}$ is obtained. We also shown that the estimator $delta_{N,1}$ is dominated by the natural estimator $delta_{N,3}$. Finally, we perform a simulation study to evaluate and compare risk functions among various competing estimators of $mu_{S}$.

Abedin Haidari, Amir T. Payandeh Najafabadi, Narayanaswamy Balakrishnan.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 150--166.

Abstract:
Consider two parallel systems, say $A$ and $B$, with respective lifetimes $T_{1}$ and $T_{2}$ wherein independent component lifetimes of each system follow exponentiated generalized gamma distribution with possibly different exponential shape and scale parameters. We show here that $T_{2}$ is smaller than $T_{1}$ with respect to the usual stochastic order (reversed hazard rate order) if the vector of logarithm (the main vector) of scale parameters of System $B$ is weakly weighted majorized by that of System $A$, and if the vector of exponential shape parameters of System $A$ is unordered mojorized by that of System $B$. By means of some examples, we show that the above results can not be extended to the hazard rate and likelihood ratio orders. However, when the scale parameters of each system divide into two homogeneous groups, we verify that the usual stochastic and reversed hazard rate orders can be extended, respectively, to the hazard rate and likelihood ratio orders. The established results complete and strengthen some of the known results in the literature.

Natalia Shenkman.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 127--135.

Abstract:
While derivations of the characterization of the $d$-variate exchangeable Marshall–Olkin copula via $d$-monotone sequences relying on basic knowledge in probability theory exist in the literature, they contain a myriad of unnecessary relatively complicated computations. We revisit this issue and provide proofs where all undesired artefacts are removed, thereby exposing the simplicity of the characterization. In particular, we give an insightful analytical derivation of the monotonicity conditions based on the monotonicity properties of the survival probabilities.

Ahmad Younso.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 112--126.

Abstract:
We consider a new nonparametric rule of classification, inspired from the classical moving window rule, that allows for the classification of spatially dependent functional data containing some completely missing curves. We investigate the consistency of this classifier under mild conditions. The practical use of the classifier will be illustrated through simulation studies.

Durba Bhattacharya, Sourabh Bhattacharya.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 71--89.

Abstract:
Present day bio-medical research is pointing towards the fact that cognizance of gene–environment interactions along with genetic interactions may help prevent or detain the onset of many complex diseases like cardiovascular disease, cancer, type2 diabetes, autism or asthma by adjustments to lifestyle. In this regard, we propose a Bayesian semiparametric model to detect not only the roles of genes and their interactions, but also the possible influence of environmental variables on the genes in case-control studies. Our model also accounts for the unknown number of genetic sub-populations via finite mixtures composed of Dirichlet processes. An effective parallel computing methodology, developed by us harnesses the power of parallel processing technology to increase the efficiencies of our conditionally independent Gibbs sampling and Transformation based MCMC (TMCMC) methods. Applications of our model and methods to simulation studies with biologically realistic genotype datasets and a real, case-control based genotype dataset on early onset of myocardial infarction (MI) have yielded quite interesting results beside providing some insights into the differential effect of gender on MI.

Flávio B. Gonçalves, Marcos O. Prates, Victor Hugo Lachos.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 51--70.

Abstract:
In this paper, we present a novel methodology to perform Bayesian model selection in linear models with heavy-tailed distributions. We consider a finite mixture of distributions to model a latent variable where each component of the mixture corresponds to one possible model within the symmetrical class of normal independent distributions. Naturally, the Gaussian model is one of the possibilities. This allows for a simultaneous analysis based on the posterior probability of each model. Inference is performed via Markov chain Monte Carlo—a Gibbs sampler with Metropolis–Hastings steps for a class of parameters. Simulated examples highlight the advantages of this approach compared to a segregated analysis based on arbitrarily chosen model selection criteria. Examples with real data are presented and an extension to censored linear regression is introduced and discussed.

Weiping Zhang, Jiangli Wang, Fang Qian, Yu Chen.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 35--50.

Abstract:
Longitudinal zero-inflated count data are widely encountered in many fields, while modeling the correlation between measurements for the same subject is more challenge due to the lack of suitable multivariate joint distributions. This paper studies a novel mean-correlation modeling approach for longitudinal zero-inflated regression model, solving both problems of specifying joint distribution and parsimoniously modeling correlations with no constraint. The joint distribution of zero-inflated discrete longitudinal responses is modeled by a copula model whose correlation parameters are innovatively represented in hyper-spherical coordinates. To overcome the computational intractability in maximizing the full likelihood function of the model, we further propose a computationally efficient pairwise likelihood approach. We then propose separated mean and correlation regression models to model these key quantities, such modeling approach can also handle irregularly and possibly subject-specific times points. The resulting estimators are shown to be consistent and asymptotically normal. Data example and simulations support the effectiveness of the proposed approach.

Fábio P. Lima, Francisco Cribari-Neto.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 18--34.

Abstract:
We address the issue of performing testing inference in small samples in the class of beta regression models. We consider the likelihood ratio test and its standard bootstrap version. We also consider two alternative resampling-based tests. One of them uses the bootstrap test statistic replicates to numerically estimate a Bartlett correction factor that can be applied to the likelihood ratio test statistic. By doing so, we avoid estimation of quantities located in the tail of the likelihood ratio test statistic null distribution. The second alternative resampling-based test uses a fast double bootstrap scheme in which a single second level bootstrapping resample is performed for each first level bootstrap replication. It delivers accurate testing inferences at a computational cost that is considerably smaller than that of a standard double bootstrapping scheme. The Monte Carlo results we provide show that the standard likelihood ratio test tends to be quite liberal in small samples. They also show that the bootstrap tests deliver accurate testing inferences even when the sample size is quite small. An empirical application is also presented and discussed.

Nandini Kannan, Debasis Kundu.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 2--17.

Abstract:
In this article, we consider models for time-to-event data obtained from experiments in which stress levels are altered at intermediate stages during the observation period. These experiments, known as step-stress tests, belong to the larger class of accelerated tests used extensively in the reliability literature. The analysis of data from step-stress tests largely relies on the popular cumulative exposure model. However, despite its simple form, the utility of the model is limited, as it is assumed that the hazard function of the underlying distribution is discontinuous at the points at which the stress levels are changed, which may not be very reasonable. Due to this deficiency, Kannan et al. ( Journal of Applied Statistics 37 (2010b) 1625–1636) introduced the cumulative risk model, where the hazard function is continuous. In this paper, we propose a class of parametric models based on the cumulative risk model assuming the underlying population contains long-term survivors or ‘cured’ fraction. An EM algorithm to compute the maximum likelihood estimators of the unknown parameters is proposed. This research is motivated by a study on altitude decompression sickness. The performance of different parametric models will be evaluated using data from this study.

Abbas Pak, M. E. Ghitany, Mohammad Reza Mahmoudi.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 894--914.

Abstract:
This paper focuses on Bayesian estimation of the parameters and reliability function of the power Lindley distribution by using various symmetric and asymmetric loss functions. Assuming suitable priors on the parameters, Bayes estimates are derived by using squared error, linear exponential (linex) and general entropy loss functions. Since, under these loss functions, Bayes estimates of the parameters do not have closed forms we use lindley’s approximation technique to calculate the Bayes estimates. Moreover, we obtain the Bayes estimates of the parameters using a Markov Chain Monte Carlo (MCMC) method. Simulation studies are conducted in order to evaluate the performances of the proposed estimators under the considered loss functions. Finally, analysis of a real data set is presented for illustrative purposes.

Dan J. Spitzner.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 861--893.

Abstract:
This article proposes a calibration scheme for Bayesian testing that coordinates analytically-derived statistical performance considerations with expert opinion. In other words, the scheme is effective and meaningful for incorporating objective elements into subjective Bayesian inference. It explores a novel role for default priors as anchors for calibration rather than substitutes for prior knowledge. Ideas are developed for use with multiplicity adjustments in multiple-model contexts, and to address the issue of prior sensitivity of Bayes factors. Along the way, the performance properties of an existing multiplicity adjustment related to the Poisson distribution are clarified theoretically. Connections of the overall calibration scheme to the Schwarz criterion are also explored. The proposed framework is examined and illustrated on a number of existing data sets related to problems in clinical trials, forensic pattern matching, and log-linear models methodology.

Wesley Bertoli, Katiane S. Conceição, Marinho G. Andrade, Francisco Louzada.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 826--860.

Abstract:
The primary goal of this paper is to introduce the zero-modified Poisson–Lindley regression model as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros in the presence of covariates. The zero-modification is incorporated by considering that a zero-truncated process produces positive observations and consequently, the proposed model can be fitted without any previous information about the zero-modification present in a given dataset. A fully Bayesian approach based on the g-prior method has been considered for inference concerns. An intensive Monte Carlo simulation study has been conducted to evaluate the performance of the developed methodology and the maximum likelihood estimators. The proposed model was considered for the analysis of a real dataset on the number of bids received by $126$ U.S. firms between 1978–1985, and the impact of choosing different prior distributions for the regression coefficients has been studied. A sensitivity analysis to detect influential points has been performed based on the Kullback–Leibler divergence. A general comparison with some well-known regression models for discrete data has been presented.

Lucas Pereira Lopes, Vicente Garibay Cancho, Francisco Louzada.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 801--825.

Abstract:
Multivariate options are adequate tools for multi-asset risk management. The pricing models derived from the pioneer Black and Scholes method under the multivariate case consider that the asset-object prices follow a Brownian geometric motion. However, the construction of such methods imposes some unrealistic constraints on the process of fair option calculation, such as constant volatility over the maturity time and linear correlation between the assets. Therefore, this paper aims to price and analyze the fair price behavior of the call-on-max (bivariate) option considering marginal heteroscedastic models with dependence structure modeled via copulas. Concerning inference, we adopt a Bayesian perspective and computationally intensive methods based on Monte Carlo simulations via Markov Chain (MCMC). A simulation study examines the bias, and the root mean squared errors of the posterior means for the parameters. Real stocks prices of Brazilian banks illustrate the approach. For the proposed method is verified the effects of strike and dependence structure on the fair price of the option. The results show that the prices obtained by our heteroscedastic model approach and copulas differ substantially from the prices obtained by the model derived from Black and Scholes. Empirical results are presented to argue the advantages of our strategy.

Flávio B. Gonçalves, Bárbara C. C. Dias.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 782--800.

Abstract:
Educational assessment usually considers a contextual questionnaire to extract relevant information from the applicants. This may include items related to socio-economical profile as well as items to extract other characteristics potentially related to applicant’s performance in the test. A careful analysis of the questionnaires jointly with the test’s results may evidence important relations between profiles and test performance. The most coherent way to perform this task in a statistical context is to use the information from the questionnaire to help explain the variability of the abilities in a joint model-based approach. Nevertheless, the responses to the questionnaire typically present missing values which, in some cases, may be missing not at random. This paper proposes a statistical methodology to model the abilities in dichotomous IRT models using the information of the contextual questionnaires via linear regression. The proposed methodology models the missing data jointly with the all the observed data, which allows for the estimation of the former. The missing data modelling is flexible enough to allow the specification of missing not at random structures. Furthermore, even if those structures are not assumed a priori, they can be estimated from the posterior results when assuming missing (completely) at random structures a priori. Statistical inference is performed under the Bayesian paradigm via an efficient MCMC algorithm. Simulated and real examples are presented to investigate the efficiency and applicability of the proposed methodology.

Glaura C. Franco, Helio S. Migon, Marcos O. Prates.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 756--781.

Abstract:
Observation and parameter driven models are commonly used in the literature to analyse time series of counts. In this paper, we study the characteristics of a variety of models and point out the main differences and similarities among these procedures, concerning parameter estimation, model fitting and forecasting. Alternatively to the literature, all inference was performed under the Bayesian paradigm. The models are fitted with a latent AR($p$) process in the mean, which accounts for autocorrelation in the data. An extensive simulation study shows that the estimates for the covariate parameters are remarkably similar across the different models. However, estimates for autoregressive coefficients and forecasts of future values depend heavily on the underlying process which generates the data. A real data set of bankruptcy in the United States is also analysed.

Marco A. R. Ferreira.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 734--744.

Abstract:
We study the limiting behavior of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly. The intrinsic conditional autoregressive model is widely used as a prior for random effects in hierarchical models for spatial modeling. This model is defined by full conditional distributions that imply an improper joint “density” with a multivariate Gaussian kernel and a singular precision matrix. To guarantee propriety of the posterior distribution, usually at the end of each iteration of the Gibbs sampler the random effects are centered to sum to zero in what is widely known as centering on the fly. While this works well in practice, this informal computational way to recenter the random effects obscures their implied prior distribution and prevents the development of formal Bayesian procedures. Here we show that the implied prior distribution, that is, the limiting distribution of the one-at-a-time Gibbs sampler for the intrinsic conditional autoregressive model with centering on the fly is a singular Gaussian distribution with a covariance matrix that is the Moore–Penrose inverse of the precision matrix. This result has important implications for the development of formal Bayesian procedures such as reference priors and Bayes-factor-based model selection for spatial models.

Sylvia Frühwirth-Schnatter.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 706--733.

Abstract:
Finite mixture models and their extensions to Markov mixture and mixture of experts models are very popular in analysing data of various kind. A challenge for these models is choosing the number of components based on marginal likelihoods. The present paper suggests two innovative, generic bridge sampling estimators of the marginal likelihood that are based on constructing balanced importance densities from the conditional densities arising during Gibbs sampling. The full permutation bridge sampling estimator is derived from considering all possible permutations of the mixture labels for a subset of these densities. For the double random permutation bridge sampling estimator, two levels of random permutations are applied, first to permute the labels of the MCMC draws and second to randomly permute the labels of the conditional densities arising during Gibbs sampling. Various applications show very good performance of these estimators in comparison to importance and to reciprocal importance sampling estimators derived from the same importance densities.

Dani Gamerman.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 686--705.

Abstract:
Point processes are one of the most commonly encountered observation processes in Spatial Statistics. Model-based inference for them depends on the likelihood function. In the most standard setting of Poisson processes, the likelihood depends on the intensity function, and can not be computed analytically. A number of approximating techniques have been proposed to handle this difficulty. In this paper, we review recent work on exact solutions that solve this problem without resorting to approximations. The presentation concentrates more heavily on discrete time but also considers continuous time. The solutions are based on model specifications that impose smoothness constraints on the intensity function. We also review approaches to include a regression component and different ways to accommodate it while accounting for additional heterogeneity. Applications are provided to illustrate the results. Finally, we discuss possible extensions to account for discontinuities and/or jumps in the intensity function.