Stochastic gradient Langevin dynamics (SGLD) is a fundamental algorithm in stochastic optimization. Recent work by Zhang et al. (2017) presents an analysis for the hitting time of SGLD for the first and second order stationary points. The proof in Zhang et al. (2017) is a two-stage procedure through bounding the Cheeger's constant, which is rather complicated and leads to loose bounds. In this paper, using intuitions from stochastic differential equations, we provide a direct analysis for the hitting times of SGLD to the first and second order stationary points. Our analysis is straightforward. It only relies on basic linear algebra and probability theory tools. Our direct analysis also leads to tighter bounds comparing to Zhang et al. (2017) and shows the explicit dependence of the hitting time on different factors, including dimensionality, smoothness, noise strength, and step size effects. Under suitable conditions, we show that the hitting time of SGLD to first-order stationary points can be dimension-independent. Moreover, we apply our analysis to study several important online estimation problems in machine learning, including linear regression, matrix factorization, and online PCA.

The horseshoe prior is frequently employed in Bayesian analysis of high-dimensional models, and has been shown to achieve minimax optimal risk properties when the truth is sparse. While optimization-based algorithms for the extremely popular Lasso and elastic net procedures can scale to dimension in the hundreds of thousands, algorithms for the horseshoe that use Markov chain Monte Carlo (MCMC) for computation are limited to problems an order of magnitude smaller. This is due to high computational cost per step and growth of the variance of time-averaging estimators as a function of dimension. We propose two new MCMC algorithms for computation in these models that have significantly improved performance compared to existing alternatives. One of the algorithms also approximates an expensive matrix product to give orders of magnitude speedup in high-dimensional applications. We prove guarantees for the accuracy of the approximate algorithm, and show that gradually decreasing the approximation error as the chain extends results in an exact algorithm. The scalability of the algorithm is illustrated in simulations with problem size as large as $N=5,000$ observations and $p=50,000$ predictors, and an application to a genome-wide association study with $N=2,267$ and $p=98,385$. The empirical results also show that the new algorithm yields estimates with lower mean squared error, intervals with better coverage, and elucidates features of the posterior that were often missed by previous algorithms in high dimensions, including bimodality of posterior marginals indicating uncertainty about which covariates belong in the model.

This paper considers a new identifiability condition for additive noise models (ANMs) in which each variable is determined by an arbitrary Borel measurable function of its parents plus an independent error. It has been shown that ANMs are fully recoverable under some identifiability conditions, such as when all error variances are equal. However, this identifiable condition could be restrictive, and hence, this paper focuses on a relaxed identifiability condition that involves not only error variances, but also the influence of parents. This new class of identifiable ANMs does not put any constraints on the form of dependencies, or distributions of errors, and allows different error variances. It further provides a statistically consistent and computationally feasible structure learning algorithm for the identifiable ANMs based on the new identifiability condition. The proposed algorithm assumes that all relevant variables are observed, while it does not assume faithfulness or a sparse graph. Demonstrated through extensive simulated and real multivariate data is that the proposed algorithm successfully recovers directed acyclic graphs.

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.

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.

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.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 2, 203--203.

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.

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.

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.

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.

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

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.

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.

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.

Achillefs Tzioufas.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 674--684.

Abstract:
The epidemic process on a graph is considered for which infectious contacts occur at rate which depends on whether a susceptible is infected for the first time or not. We show that the Vasershtein coupling extends if and only if secondary infections occur at rate which is greater than that of initial ones. Nonetheless we show that, with respect to the probability of occurrence of an infinite epidemic, the said proviso may be dropped regarding the totally asymmetric process in one dimension, thus settling in the affirmative this special case of the conjecture for arbitrary graphs due to [ Ann. Appl. Probab. 13 (2003) 669–690].

Kazuki Okamura.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 586--637.

Abstract:
We consider a random object that is associated with both random walks and random media, specifically, the superposition of a configuration of subcritical Bernoulli percolation on an infinite connected graph and the trace of the simple random walk on the same graph. We investigate asymptotics for the number of vertices of the enlargement of the trace of the walk until a fixed time, when the time tends to infinity. This process is more highly self-interacting than the range of random walk, which yields difficulties. We show a law of large numbers on vertex-transitive transient graphs. We compare the process on a vertex-transitive graph with the process on a finitely modified graph of the original vertex-transitive graph and show their behaviors are similar. We show that the process fluctuates almost surely on a certain non-vertex-transitive graph. On the two-dimensional integer lattice, by investigating the size of the boundary of the trace, we give an estimate for variances of the process implying a law of large numbers. We give an example of a graph with unbounded degrees on which the process behaves in a singular manner. As by-products, some results for the range and the boundary, which will be of independent interest, are obtained.

Davide Gabrielli, Ida Germana Minelli.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 558--585.

Abstract:
Stochastic monotonicity is a well-known partial order relation between probability measures defined on the same partially ordered set. Strassen theorem establishes equivalence between stochastic monotonicity and the existence of a coupling compatible with respect to the partial order. We consider the case of a countable set and introduce the class of finitely decomposable flows on a directed acyclic graph associated to the partial order. We show that a probability measure stochastically dominates another probability measure if and only if there exists a finitely decomposable flow having divergence given by the difference of the two measures. We illustrate the result with some examples.

Christian Rau.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 549--557.

Abstract:
We revisit a shape inversion formula derived by Panaretos in the context of a particle density estimation problem with unknown rotation of the particle. A distribution is presented which imitates, or “fakes”, the uniformity or Haar distribution that is part of that formula.

Christian Olivera, Ciprian Tudor.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 520--531.

Abstract:
Via a special transform and by using the techniques of the Malliavin calculus, we analyze the density of the solution to a stochastic differential equation with unbounded drift.

Katarzyna Jańczak-Borkowska.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 480--497.

Abstract:
We prove the existence and uniqueness of the solution of backward stochastic variational inequalities with respect to fractional Brownian motion and with non-Lipschitz coefficient. We assume that $H>1/2$.

Rosineide F. da Paz, Narayanaswamy Balakrishnan, Jorge Luis Bazán.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 455--479.

Abstract:
Tadikamalla and Johnson [ Biometrika 69 (1982) 461–465] developed the $L_{B}$ distribution to variables with bounded support by considering a transformation of the standard Logistic distribution. In this manuscript, a convenient parametrization of this distribution is proposed in order to develop regression models. This distribution, referred to here as L-Logistic distribution, provides great flexibility and includes the uniform distribution as a particular case. Several properties of this distribution are studied, and a Bayesian approach is adopted for the parameter estimation. Simulation studies, considering prior sensitivity analysis, recovery of parameters and comparison of algorithms, and robustness to outliers are all discussed showing that the results are insensitive to the choice of priors, efficiency of the algorithm MCMC adopted, and robustness of the model when compared with the beta distribution. Applications to estimate the vulnerability to poverty and to explain the anxiety are performed. The results to applications show that the L-Logistic regression models provide a better fit than the corresponding beta regression models.

Daniel Ahlberg.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 397--401.

Abstract:
We study the rate of convergence in the celebrated Shape Theorem in first-passage percolation, obtaining the precise asymptotic rate of decay for the probability of linear order deviations under a moment condition. Our results are presented from a temporal perspective and complement previous work by the same author, in which the rate of convergence was studied from the standard spatial perspective.

Rodrigo Citton P. dos Reis, Enrico A. Colosimo, Gustavo L. Gilardoni.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 374--396.

Abstract:
In the data analysis from multiple repairable systems, it is usual to observe both different truncation times and heterogeneity among the systems. Among other reasons, the latter is caused by different manufacturing lines and maintenance teams of the systems. In this paper, a hierarchical model is proposed for the statistical analysis of multiple repairable systems under different truncation times. A reparameterization of the power law process is proposed in order to obtain a quasi-conjugate bayesian analysis. An empirical Bayes approach is used to estimate model hyperparameters. The uncertainty in the estimate of these quantities are corrected by using a parametric bootstrap approach. The results are illustrated in a real data set of failure times of power transformers from an electric company in Brazil.

Bastien Mallein.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 356--373.

Abstract:
Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $lambda^{*}n^{1/3}$ for some explicit constant $lambda^{*}$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.

Francisco Cribari-Neto, Rodney V. Fonseca.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 329--355.

Abstract:
The log-linear Birnbaum–Saunders model has been widely used in empirical applications. We introduce an extension of this model based on a recently proposed version of the Birnbaum–Saunders distribution which is more flexible than the standard Birnbaum–Saunders law since its density may assume both unimodal and bimodal shapes. We show how to perform point estimation, interval estimation and hypothesis testing inferences on the parameters that index the regression model we propose. We also present a number of diagnostic tools, such as residual analysis, local influence, generalized leverage, generalized Cook’s distance and model misspecification tests. We investigate the usefulness of model selection criteria and the accuracy of prediction intervals for the proposed model. Results of Monte Carlo simulations are presented. Finally, we also present and discuss an empirical application.

Khamis K. Said, Wei Ning, Yubin Tian.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 280--300.

Abstract:
In this paper, we study the change point problem for the skew normal distribution model from the view of model selection problem. The detection procedure based on the modified information criterion (MIC) for change problem is proposed. Such a procedure has advantage in detecting the changes in early and late stage of a data comparing to the one based on the traditional Schwarz information criterion which is well known as Bayesian information criterion (BIC) by considering the complexity of the models. Due to the difficulty in deriving the analytic asymptotic distribution of the test statistic based on the MIC procedure, the bootstrap simulation is provided to obtain the critical values at the different significance levels. Simulations are conducted to illustrate the comparisons of performance between MIC, BIC and likelihood ratio test (LRT). Such an approach is applied on two stock market data sets to indicate the detection procedure.

Qianying Liu, Clare A. McGrory, Peter W. J. Baxter.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 267--279.

Abstract:
The pressing need for improved methods for analysing and coping with big data has opened up a new area of research for statisticians. Image analysis is an area where there is typically a very large number of data points to be processed per image, and often multiple images are captured over time. These issues make it challenging to design methodology that is reliable and yet still efficient enough to be of practical use. One promising emerging approach for this problem is to reduce the amount of data that actually has to be processed by extracting what we call coresets from the full dataset; analysis is then based on the coreset rather than the whole dataset. Coresets are representative subsamples of data that are carefully selected via an adaptive sampling approach. We propose a new approach called coreset variational Bayes (CVB) for mixture modelling; this is an algorithm which can perform a variational Bayes analysis of a dataset based on just an extracted coreset of the data. We apply our algorithm to weed image analysis.

Kushal K. Dey, Sourabh Bhattacharya.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 222--266.

Abstract:
Transformation based Markov Chain Monte Carlo (TMCMC) was proposed by Dutta and Bhattacharya ( Statistical Methodology 16 (2014) 100–116) as an efficient alternative to the Metropolis–Hastings algorithm, especially in high dimensions. The main advantage of this algorithm is that it simultaneously updates all components of a high dimensional parameter using appropriate move types defined by deterministic transformation of a single random variable. This results in reduction in time complexity at each step of the chain and enhances the acceptance rate. In this paper, we first provide a brief review of the optimal scaling theory for various existing MCMC approaches, comparing and contrasting them with the corresponding TMCMC approaches.The optimal scaling of the simplest form of TMCMC, namely additive TMCMC , has been studied extensively for the Gaussian proposal density in Dey and Bhattacharya (2017a). Here, we discuss diffusion-based optimal scaling behavior of additive TMCMC for non-Gaussian proposal densities—in particular, uniform, Student’s $t$ and Cauchy proposals. Although we could not formally prove our diffusion result for the Cauchy proposal, simulation based results lead us to conjecture that at least the recipe for obtaining general optimal scaling and optimal acceptance rate holds for the Cauchy case as well. We also consider diffusion based optimal scaling of TMCMC when the target density is discontinuous. Such non-regular situations have been studied in the case of Random Walk Metropolis Hastings (RWMH) algorithm by Neal and Roberts ( Methodology and Computing in Applied Probability 13 (2011) 583–601) using expected squared jumping distance (ESJD), but the diffusion theory based scaling has not been considered. We compare our diffusion based optimally scaled TMCMC approach with the ESJD based optimally scaled RWM with simulation studies involving several target distributions and proposal distributions including the challenging Cauchy proposal case, showing that additive TMCMC outperforms RWMH in almost all cases considered.

Ivana Ilić, Vladica M. Veličković.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 192--203.

Abstract:
Financial returns are known to be nonnormal and tend to have fat-tailed distribution. Also, the dependence of large values in a stochastic process is an important topic in risk, insurance and finance. In the presence of missing values, we deal with the asymptotic properties of a simple “median” estimator of the tail index based on random variables with the heavy-tailed distribution function and certain dependence among the extremes. Weak consistency and asymptotic normality of the proposed estimator are established. The estimator is a special case of a well-known estimator defined in Bacro and Brito [ Statistics & Decisions 3 (1993) 133–143]. The advantage of the estimator is its robustness against deviations and compared to Hill’s, it is less affected by the fluctuations related to the maximum of the sample or by the presence of outliers. Several examples are analyzed in order to support the proofs.

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.