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.

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.

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.

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.

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.

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.

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.

Hedibert F. Lopes, Nicholas G. Polson.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 745--755.

Abstract:
Bayesian hypothesis testing is re-examined from the perspective of an a priori assessment of the test statistic distribution under the alternative. By assessing the distribution of an observable test statistic, rather than prior parameter values, we revisit the seminal paper of Edwards, Lindman and Savage ( Psychol. Rev. 70 (1963) 193–242). There are a number of important take-aways from comparing the Bayesian paradigm via Bayes factors to frequentist ones. We provide examples where evidence for a Bayesian strikingly supports the null, but leads to rejection under a classical test. Finally, we conclude with directions for future research.

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.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 4, 685--685.

Yair Y. Shaki.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 532--548.

Abstract:
We consider a Jackson network in a general heavy traffic diffusion regime with the $alpha$-parametrization . We also assume that each customer may abandon the system while waiting. We show that in this regime the queue-length process converges to a multi-dimensional regulated Ornstein–Uhlenbeck process.

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.

Salem M. Al-Gezeri, Robert G. Aykroyd.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 3, 498--519.

Abstract:
The use of Markov random field (MRF) models has proven to be a fruitful approach in a wide range of image processing applications. It allows local texture information to be incorporated in a systematic and unified way and allows statistical inference theory to be applied giving rise to novel output summaries and enhanced image interpretation. A great advantage of such low-level approaches is that they lead to flexible models, which can be applied to a wide range of imaging problems without the need for significant modification. This paper proposes and explores the use of conditional MRF models for situations where multiple images are to be processed simultaneously, or where only a single image is to be reconstructed and a sequential approach is taken. Although the coupling of image intensity values is a special case of our approach, the main extension over previous proposals is to allow the direct coupling of other properties, such as smoothness or texture. This is achieved using a local modulating function which adjusts the influence of global smoothing without the need for a fully inhomogeneous prior model. Several modulating functions are considered and a detailed simulation study, motivated by remote sensing applications in archaeological geophysics, of conditional reconstruction is presented. The results demonstrate that a substantial improvement in the quality of the image reconstruction, in terms of errors and residuals, can be achieved using this approach, especially at locations with rapid changes in the underlying intensity.

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.

Luisa Rivas, Manuel Galea.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 402--424.

Abstract:
In this paper, we present a regression model where the response variable is a count data that follows a Waring distribution. The Waring regression model allows for analysis of phenomena where the Geometric regression model is inadequate, because the probability of success on each trial, $p$, is different for each individual and $p$ has an associated distribution. Estimation is performed by maximum likelihood, through the maximization of the $Q$-function using EM algorithm. Diagnostic measures are calculated for this model. To illustrate the results, an application to real data is presented. Some specific details are given in the Appendix of the paper.

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.

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.

Emilia Athayde, Assis Azevedo, Michelli Barros, Víctor Leiva.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 301--328.

Abstract:
The Birnbaum–Saunders (BS) distribution has been largely studied and applied. A random variable with BS distribution is a transformation of another random variable with standard normal distribution. Generalized BS distributions are obtained when the normally distributed random variable is replaced by another symmetrically distributed random variable. This allows us to obtain a wide class of positively skewed models with lighter and heavier tails than the BS model. Its failure rate admits several shapes, including the unimodal case, with its change-point being able to be used for different purposes. For example, to establish the reduction in a dose, and then in the cost of the medical treatment. We analyze the failure rates of generalized BS distributions obtained by the logistic, normal and Student-t distributions, considering their shape and change-point, estimating them, evaluating their robustness, assessing their performance by simulations, and applying the results to real data from different areas.

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.

Alain Desgagné, Philippe Gagnon.

Source: Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 205--221.

Abstract:
Whole robustness is a nice property to have for statistical models. It implies that the impact of outliers gradually vanishes as they approach plus or minus infinity. So far, the Bayesian literature provides results that ensure whole robustness for the location-scale model. In this paper, we make two contributions. First, we generalise the results to attain whole robustness in simple linear regression through the origin, which is a necessary step towards results for general linear regression models. We allow the variance of the error term to depend on the explanatory variable. This flexibility leads to the second contribution: we provide a simple Bayesian approach to robustly estimate finite population means and ratios. The strategy to attain whole robustness is simple since it lies in replacing the traditional normal assumption on the error term by a super heavy-tailed distribution assumption. As a result, users can estimate the parameters as usual, using the posterior distribution.

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.

Norbert Hirschauer, Sven Grüner, Oliver Mußhoff, Claudia Becker, Antje Jantsch.

Source: Statistics Surveys, Volume 14, 71--91.

Abstract:
Besides the inferential errors that abound in the interpretation of $p$-values, the probabilistic pre-conditions (i.e. random sampling or equivalent) for using them at all are not often met by observational studies in the social sciences. This paper systematizes different sampling designs and discusses the restrictive requirements of data collection that are the indispensable prerequisite for using $p$-values.

John J. Dziak, Donna L. Coffman, Matthew Reimherr, Justin Petrovich, Runze Li, Saul Shiffman, Mariya P. Shiyko.

Source: Statistics Surveys, Volume 13, 150--180.

Abstract:
Researchers are sometimes interested in predicting a distal or external outcome (such as smoking cessation at follow-up) from the trajectory of an intensively recorded longitudinal variable (such as urge to smoke). This can be done in a semiparametric way via scalar-on-function regression. However, the resulting fitted coefficient regression function requires special care for correct interpretation, as it represents the joint relationship of time points to the outcome, rather than a marginal or cross-sectional relationship. We provide practical guidelines, based on experience with scientific applications, for helping practitioners interpret their results and illustrate these ideas using data from a smoking cessation study.

Pierre Lafaye de Micheaux, Benoît Liquet, Matthew Sutton.

Source: Statistics Surveys, Volume 13, 119--149.

Abstract:
Partial Least Squares (PLS) methods have been heavily exploited to analyse the association between two blocks of data. These powerful approaches can be applied to data sets where the number of variables is greater than the number of observations and in the presence of high collinearity between variables. Different sparse versions of PLS have been developed to integrate multiple data sets while simultaneously selecting the contributing variables. Sparse modeling is a key factor in obtaining better estimators and identifying associations between multiple data sets. The cornerstone of the sparse PLS methods is the link between the singular value decomposition (SVD) of a matrix (constructed from deflated versions of the original data) and least squares minimization in linear regression. We review four popular PLS methods for two blocks of data. A unified algorithm is proposed to perform all four types of PLS including their regularised versions. We present various approaches to decrease the computation time and show how the whole procedure can be scalable to big data sets. The bigsgPLS R package implements our unified algorithm and is available at https://github.com/matt-sutton/bigsgPLS .

Solveig Engebretsen, Ingrid K. Glad.

Source: Statistics Surveys, Volume 13, 1--51.

Abstract:
In numerous problems where the aim is to estimate the effect of a predictor variable on a response, one can assume a monotone relationship. For example, dose-effect models in medicine are of this type. In a multiple regression setting, additive monotone regression models assume that each predictor has a monotone effect on the response. In this paper, we present an overview and comparison of very recent frequentist methods for fitting additive monotone regression models. Three of the methods we present can be used both in the high dimensional setting, where the number of parameters $p$ exceeds the number of observations $n$, and in the classical multiple setting where $1<pleq n$. However, many of the most recent methods only apply to the classical setting. The methods are compared through simulation experiments in terms of efficiency, prediction error and variable selection properties in both settings, and they are applied to the Boston housing data. We conclude with some recommendations on when the various methods perform best.

Bomin Kim, Kevin H. Lee, Lingzhou Xue, Xiaoyue Niu.

Source: Statistics Surveys, Volume 12, 105--135.

Abstract:
We present a selective review of statistical modeling of dynamic networks. We focus on models with latent variables, specifically, the latent space models and the latent class models (or stochastic blockmodels), which investigate both the observed features and the unobserved structure of networks. We begin with an overview of the static models, and then we introduce the dynamic extensions. For each dynamic model, we also discuss its applications that have been studied in the literature, with the data source listed in Appendix. Based on the review, we summarize a list of open problems and challenges in dynamic network modeling with latent variables.

Phillip S. Kott.

Source: Statistics Surveys, Volume 12, 1--17.

Abstract:
Fitting complex survey data to regression equations is explored under a design-sensitive model-based framework. A robust version of the standard model assumes that the expected value of the difference between the dependent variable and its model-based prediction is zero no matter what the values of the explanatory variables. The extended model assumes only that the difference is uncorrelated with the covariates. Little is assumed about the error structure of this difference under either model other than independence across primary sampling units. The standard model often fails in practice, but the extended model very rarely does. Under this framework some of the methods developed in the conventional design-based, pseudo-maximum-likelihood framework, such as fitting weighted estimating equations and sandwich mean-squared-error estimation, are retained but their interpretations change. Few of the ideas here are new to the refereed literature. The goal instead is to collect those ideas and put them into a unified conceptual framework.

Jonathan R. Bradley, Noel Cressie, Tao Shi.

Source: Statistics Surveys, Volume 10, 100--131.

Abstract:
In this article, we review and compare a number of methods of spatial prediction, where each method is viewed as an algorithm that processes spatial data. To demonstrate the breadth of available choices, we consider both traditional and more-recently-introduced spatial predictors. Specifically, in our exposition we review: traditional stationary kriging, smoothing splines, negative-exponential distance-weighting, fixed rank kriging, modified predictive processes, a stochastic partial differential equation approach, and lattice kriging. This comparison is meant to provide a service to practitioners wishing to decide between spatial predictors. Hence, we provide technical material for the unfamiliar, which includes the definition and motivation for each (deterministic and stochastic) spatial predictor. We use a benchmark dataset of $mathrm{CO}_{2}$ data from NASA’s AIRS instrument to address computational efficiencies that include CPU time and memory usage. Furthermore, the predictive performance of each spatial predictor is assessed empirically using a hold-out subset of the AIRS data.

Mariella Dimiccoli.

Source: Statistics Surveys, Volume 10, 53--99.

Abstract:
Cone regression is a particular case of quadratic programming that minimizes a weighted sum of squared residuals under a set of linear inequality constraints. Several important statistical problems such as isotonic, concave regression or ANOVA under partial orderings, just to name a few, can be considered as particular instances of the cone regression problem. Given its relevance in Statistics, this paper aims to address the fundamentals of cone regression from a theoretical and practical point of view. Several formulations of the cone regression problem are considered and, focusing on the particular case of concave regression as an example, several algorithms are analyzed and compared both qualitatively and quantitatively through numerical simulations. Several improvements to enhance numerical stability and bound the computational cost are proposed. For each analyzed algorithm, the pseudo-code and its corresponding code in Matlab are provided. The results from this study demonstrate that the choice of the optimization approach strongly impacts the numerical performances. It is also shown that methods are not currently available to solve efficiently cone regression problems with large dimension (more than many thousands of points). We suggest further research to fill this gap by exploiting and adapting classical multi-scale strategy to compute an approximate solution.

Clément Marteau, Theofanis Sapatinas.

Source: Statistics Surveys, Volume 9, 253--297.

Abstract:
We are concerned with minimax signal detection. In this setting, we discuss non-asymptotic and asymptotic approaches through a unified treatment. In particular, we consider a Gaussian sequence model that contains classical models as special cases, such as, direct, well-posed inverse and ill-posed inverse problems. Working with certain ellipsoids in the space of squared-summable sequences of real numbers, with a ball of positive radius removed, we compare the construction of lower and upper bounds for the minimax separation radius (non-asymptotic approach) and the minimax separation rate (asymptotic approach) that have been proposed in the literature. Some additional contributions, bringing to light links between non-asymptotic and asymptotic approaches to minimax signal, are also presented. An example of a mildly ill-posed inverse problem is used for illustrative purposes. In particular, it is shown that tools used to derive ‘asymptotic’ results can be exploited to draw ‘non-asymptotic’ conclusions, and vice-versa. In order to enhance our understanding of these two minimax signal detection paradigms, we bring into light hitherto unknown similarities and links between non-asymptotic and asymptotic approaches.

Kevin McGoff, Sayan Mukherjee, Natesh Pillai.

Source: Statistics Surveys, Volume 9, 209--252.

Abstract:
The topic of statistical inference for dynamical systems has been studied widely across several fields. In this survey we focus on methods related to parameter estimation for nonlinear dynamical systems. Our objective is to place results across distinct disciplines in a common setting and highlight opportunities for further research.

Didier Chauveau, David R. Hunter, Michael Levine.

Source: Statistics Surveys, Volume 9, 1--31.

Abstract:
The conditional independence assumption for nonparametric multivariate finite mixture models, a weaker form of the well-known conditional independence assumption for random effects models for longitudinal data, is the subject of an increasing number of theoretical and algorithmic developments in the statistical literature. After presenting a survey of this literature, including an in-depth discussion of the all-important identifiability results, this article describes and extends an algorithm for estimation of the parameters in these models. The algorithm works for any number of components in three or more dimensions. It possesses a descent property and can be easily adapted to situations where the data are grouped in blocks of conditionally independent variables. We discuss how to adapt this algorithm to various location-scale models that link component densities, and we even adapt it to a particular class of univariate mixture problems in which the components are assumed symmetric. We give a bandwidth selection procedure for our algorithm. Finally, we demonstrate the effectiveness of our algorithm using a simulation study and two psychometric datasets.

Adrien Saumard, Jon A. Wellner.

Source: Statistics Surveys, Volume 8, 45--114.

Abstract:
We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strong log-concavity on $mathbb{R}$ under convolution follows from a fundamental monotonicity result of Efron (1965). We provide a new proof of Efron’s theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Aki Vehtari, Janne Ojanen.

Source: Statistics Surveys, Volume 8, , 1--1.

Abstract:
Errata for “A survey of Bayesian predictive methods for model assessment, selection and comparison” by A. Vehtari and J. Ojanen, Statistics Surveys , 6 (2012), 142–228. doi:10.1214/12-SS102.

Aki Vehtari, Janne Ojanen

Source: Statist. Surv., Volume 6, 142--228.

Abstract:
To date, several methods exist in the statistical literature for model assessment, which purport themselves specifically as Bayesian predictive methods. The decision theoretic assumptions on which these methods are based are not always clearly stated in the original articles, however. The aim of this survey is to provide a unified review of Bayesian predictive model assessment and selection methods, and of methods closely related to them. We review the various assumptions that are made in this context and discuss the connections between different approaches, with an emphasis on how each method approximates the expected utility of using a Bayesian model for the purpose of predicting future data.