We consider the problem of jointly estimating multiple inverse covariance matrices from high-dimensional data consisting of distinct classes. An $ell_2$-penalized maximum likelihood approach is employed. The suggested approach is flexible and generic, incorporating several other $ell_2$-penalized estimators as special cases. In addition, the approach allows specification of target matrices through which prior knowledge may be incorporated and which can stabilize the estimation procedure in high-dimensional settings. The result is a targeted fused ridge estimator that is of use when the precision matrices of the constituent classes are believed to chiefly share the same structure while potentially differing in a number of locations of interest. It has many applications in (multi)factorial study designs. We focus on the graphical interpretation of precision matrices with the proposed estimator then serving as a basis for integrative or meta-analytic Gaussian graphical modeling. Situations are considered in which the classes are defined by data sets and subtypes of diseases. The performance of the proposed estimator in the graphical modeling setting is assessed through extensive simulation experiments. Its practical usability is illustrated by the differential network modeling of 12 large-scale gene expression data sets of diffuse large B-cell lymphoma subtypes. The estimator and its related procedures are incorporated into the R-package rags2ridges.

This paper considers a Bayesian approach to graph-based semi-supervised learning. We show that if the graph parameters are suitably scaled, the graph-posteriors converge to a continuum limit as the size of the unlabeled data set grows. This consistency result has profound algorithmic implications: we prove that when consistency holds, carefully designed Markov chain Monte Carlo algorithms have a uniform spectral gap, independent of the number of unlabeled inputs. Numerical experiments illustrate and complement the theory.

Sufficient dimension reduction (SDR) is a very useful concept for exploratory analysis and data visualization in regression, especially when the number of covariates is large. Many SDR methods have been proposed for regression with a continuous response, where the central subspace (CS) is the target of estimation. Various conditions, such as the linearity condition and the constant covariance condition, are imposed so that these methods can estimate at least a portion of the CS. In this paper we study SDR for regression and discriminant analysis with categorical response. Motivated by the exploratory analysis and data visualization aspects of SDR, we propose a new geometric framework to reformulate the SDR problem in terms of manifold optimization and introduce a new concept called Maximum Separation Subspace (MASES). The MASES naturally preserves the “sufficiency” in SDR without imposing additional conditions on the predictor distribution, and directly inspires a semi-parametric estimator. Numerical studies show MASES exhibits superior performance as compared with competing SDR methods in specific settings.

Consider an unknown smooth function $f: [0,1]^d ightarrow mathbb{R}$, and assume we are given $n$ noisy mod 1 samples of $f$, i.e., $y_i = (f(x_i) + eta_i) mod 1$, for $x_i in [0,1]^d$, where $eta_i$ denotes the noise. Given the samples $(x_i,y_i)_{i=1}^{n}$, our goal is to recover smooth, robust estimates of the clean samples $f(x_i) mod 1$. We formulate a natural approach for solving this problem, which works with angular embeddings of the noisy mod 1 samples over the unit circle, inspired by the angular synchronization framework. This amounts to solving a smoothness regularized least-squares problem -- a quadratically constrained quadratic program (QCQP) -- where the variables are constrained to lie on the unit circle. Our proposed approach is based on solving its relaxation, which is a trust-region sub-problem and hence solvable efficiently. We provide theoretical guarantees demonstrating its robustness to noise for adversarial, as well as random Gaussian and Bernoulli noise models. To the best of our knowledge, these are the first such theoretical results for this problem. We demonstrate the robustness and efficiency of our proposed approach via extensive numerical simulations on synthetic data, along with a simple least-squares based solution for the unwrapping stage, that recovers the original samples of $f$ (up to a global shift). It is shown to perform well at high levels of noise, when taking as input the denoised modulo $1$ samples. Finally, we also consider two other approaches for denoising the modulo 1 samples that leverage tools from Riemannian optimization on manifolds, including a Burer-Monteiro approach for a semidefinite programming relaxation of our formulation. For the two-dimensional version of the problem, which has applications in synthetic aperture radar interferometry (InSAR), we are able to solve instances of real-world data with a million sample points in under 10 seconds, on a personal laptop.

We propose graph-dependent implicit regularisation strategies for synchronised distributed stochastic subgradient descent (Distributed SGD) for convex problems in multi-agent learning. Under the standard assumptions of convexity, Lipschitz continuity, and smoothness, we establish statistical learning rates that retain, up to logarithmic terms, single-machine serial statistical guarantees through implicit regularisation (step size tuning and early stopping) with appropriate dependence on the graph topology. Our approach avoids the need for explicit regularisation in decentralised learning problems, such as adding constraints to the empirical risk minimisation rule. Particularly for distributed methods, the use of implicit regularisation allows the algorithm to remain simple, without projections or dual methods. To prove our results, we establish graph-independent generalisation bounds for Distributed SGD that match the single-machine serial SGD setting (using algorithmic stability), and we establish graph-dependent optimisation bounds that are of independent interest. We present numerical experiments to show that the qualitative nature of the upper bounds we derive can be representative of real behaviours.

Over the past decades, numerous loss functions have been been proposed for a variety of supervised learning tasks, including regression, classification, ranking, and more generally structured prediction. Understanding the core principles and theoretical properties underpinning these losses is key to choose the right loss for the right problem, as well as to create new losses which combine their strengths. In this paper, we introduce Fenchel-Young losses, a generic way to construct a convex loss function for a regularized prediction function. We provide an in-depth study of their properties in a very broad setting, covering all the aforementioned supervised learning tasks, and revealing new connections between sparsity, generalized entropies, and separation margins. We show that Fenchel-Young losses unify many well-known loss functions and allow to create useful new ones easily. Finally, we derive efficient predictive and training algorithms, making Fenchel-Young losses appealing both in theory and practice.

We consider the problem of learning causal models from observational data generated by linear non-Gaussian acyclic causal models with latent variables. Without considering the effect of latent variables, the inferred causal relationships among the observed variables are often wrong. Under faithfulness assumption, we propose a method to check whether there exists a causal path between any two observed variables. From this information, we can obtain the causal order among the observed variables. The next question is whether the causal effects can be uniquely identified as well. We show that causal effects among observed variables cannot be identified uniquely under mere assumptions of faithfulness and non-Gaussianity of exogenous noises. However, we are able to propose an efficient method that identifies the set of all possible causal effects that are compatible with the observational data. We present additional structural conditions on the causal graph under which causal effects among observed variables can be determined uniquely. Furthermore, we provide necessary and sufficient graphical conditions for unique identification of the number of variables in the system. Experiments on synthetic data and real-world data show the effectiveness of our proposed algorithm for learning causal models.

Given a response $Y$ and a vector $X = (X^1, dots, X^d)$ of $d$ predictors, we investigate the problem of inferring direct causes of $Y$ among the vector $X$. Models for $Y$ that use all of its causal covariates as predictors enjoy the property of being invariant across different environments or interventional settings. Given data from such environments, this property has been exploited for causal discovery. Here, we extend this inference principle to situations in which some (discrete-valued) direct causes of $ Y $ are unobserved. Such cases naturally give rise to switching regression models. We provide sufficient conditions for the existence, consistency and asymptotic normality of the MLE in linear switching regression models with Gaussian noise, and construct a test for the equality of such models. These results allow us to prove that the proposed causal discovery method obtains asymptotic false discovery control under mild conditions. We provide an algorithm, make available code, and test our method on simulated data. It is robust against model violations and outperforms state-of-the-art approaches. We further apply our method to a real data set, where we show that it does not only output causal predictors, but also a process-based clustering of data points, which could be of additional interest to practitioners.

We present a probabilistic framework for studying adversarial attacks on discrete data. Based on this framework, we derive a perturbation-based method, Greedy Attack, and a scalable learning-based method, Gumbel Attack, that illustrate various tradeoffs in the design of attacks. We demonstrate the effectiveness of these methods using both quantitative metrics and human evaluation on various state-of-the-art models for text classification, including a word-based CNN, a character-based CNN and an LSTM. As an example of our results, we show that the accuracy of character-based convolutional networks drops to the level of random selection by modifying only five characters through Greedy Attack.

Parametrised state space models in the form of recurrent networks are often used in machine learning to learn from data streams exhibiting temporal dependencies. To break the black box nature of such models it is important to understand the dynamical features of the input-driving time series that are formed in the state space. We propose a framework for rigorous analysis of such state representations in vanishing memory state space models such as echo state networks (ESN). In particular, we consider the state space a temporal feature space and the readout mapping from the state space a kernel machine operating in that feature space. We show that: (1) The usual ESN strategy of randomly generating input-to-state, as well as state coupling leads to shallow memory time series representations, corresponding to cross-correlation operator with fast exponentially decaying coefficients; (2) Imposing symmetry on dynamic coupling yields a constrained dynamic kernel matching the input time series with straightforward exponentially decaying motifs or exponentially decaying motifs of the highest frequency; (3) Simple ring (cycle) high-dimensional reservoir topology specified only through two free parameters can implement deep memory dynamic kernels with a rich variety of matching motifs. We quantify richness of feature representations imposed by dynamic kernels and demonstrate that for dynamic kernel associated with cycle reservoir topology, the kernel richness undergoes a phase transition close to the edge of stability.

pyts is an open-source Python package for time series classification. This versatile toolbox provides implementations of many algorithms published in the literature, preprocessing functionalities, and data set loading utilities. pyts relies on the standard scientific Python packages numpy, scipy, scikit-learn, joblib, and numba, and is distributed under the BSD-3-Clause license. Documentation contains installation instructions, a detailed user guide, a full API description, and concrete self-contained examples.

We develop ancestral Gumbel-Top-$k$ sampling: a generic and efficient method for sampling without replacement from discrete-valued Bayesian networks, which includes multivariate discrete distributions, Markov chains and sequence models. The method uses an extension of the Gumbel-Max trick to sample without replacement by finding the top $k$ of perturbed log-probabilities among all possible configurations of a Bayesian network. Despite the exponentially large domain, the algorithm has a complexity linear in the number of variables and sample size $k$. Our algorithm allows to set the number of parallel processors $m$, to trade off the number of iterations versus the total cost (iterations times $m$) of running the algorithm. For $m = 1$ the algorithm has minimum total cost, whereas for $m = k$ the number of iterations is minimized, and the resulting algorithm is known as Stochastic Beam Search. We provide extensions of the algorithm and discuss a number of related algorithms. We analyze the properties of ancestral Gumbel-Top-$k$ sampling and compare against alternatives on randomly generated Bayesian networks with different levels of connectivity. In the context of (deep) sequence models, we show its use as a method to generate diverse but high-quality translations and statistical estimates of translation quality and entropy.

We consider the skill rating problem for multiplayer games, that is how to infer player skills from game outcomes in multiplayer games. We formulate the problem as a minimization problem $arg min_{s} s^T Delta s$ where $Delta$ is a positive semidefinite matrix and $s$ a real-valued function, of which some entries are the skill values to be inferred and other entries are constrained by the game outcomes. We leverage graph-based semi-supervised learning (SSL) algorithms for this problem. We apply our algorithms on several data sets of multiplayer games and obtain very promising results compared to Elo Duelling (see Elo, 1978) and TrueSkill (see Herbrich et al., 2006).. As we leverage graph-based SSL algorithms and because games can be seen as relations between sets of players, we then generalize the approach. For this aim, we introduce a new finite model, called hypernode graph, defined to be a set of weighted binary relations between sets of nodes. We define Laplacians of hypernode graphs. Then, we show that the skill rating problem for multiplayer games can be formulated as $arg min_{s} s^T Delta s$ where $Delta$ is the Laplacian of a hypernode graph constructed from a set of games. From a fundamental perspective, we show that hypernode graph Laplacians are symmetric positive semidefinite matrices with constant functions in their null space. We show that problems on hypernode graphs can not be solved with graph constructions and graph kernels. We relate hypernode graphs to signed graphs showing that positive relations between groups can lead to negative relations between individuals.

We consider the problem of unveiling the implicit network structure of node interactions (such as user interactions in a social network), based only on high-frequency timestamps. Our inference is based on the minimization of the least-squares loss associated with a multivariate Hawkes model, penalized by $ell_1$ and trace norm of the interaction tensor. We provide a first theoretical analysis for this problem, that includes sparsity and low-rank inducing penalizations. This result involves a new data-driven concentration inequality for matrix martingales in continuous time with observable variance, which is a result of independent interest and a broad range of possible applications since it extends to matrix martingales former results restricted to the scalar case. A consequence of our analysis is the construction of sharply tuned $ell_1$ and trace-norm penalizations, that leads to a data-driven scaling of the variability of information available for each users. Numerical experiments illustrate the significant improvements achieved by the use of such data-driven penalizations.

In this paper we introduce a statistical model, called additively faithful directed acyclic graph (AFDAG), for causal learning from observational data. Our approach is based on additive conditional independence (ACI), a recently proposed three-way statistical relation that shares many similarities with conditional independence but without resorting to multi-dimensional kernels. This distinct feature strikes a balance between a parametric model and a fully nonparametric model, which makes the proposed model attractive for handling large networks. We develop an estimator for AFDAG based on a linear operator that characterizes ACI, and establish the consistency and convergence rates of this estimator, as well as the uniform consistency of the estimated DAG. Moreover, we introduce a modified PC-algorithm to implement the estimating procedure efficiently, so that its complexity is determined by the level of sparseness rather than the dimension of the network. Through simulation studies we show that our method outperforms existing methods when commonly assumed conditions such as Gaussian or Gaussian copula distributions do not hold. Finally, the usefulness of AFDAG formulation is demonstrated through an application to a proteomics data set.

Regularized least-squares (kernel-ridge / Gaussian process) regression is a fundamental algorithm of statistics and machine learning. Because generic algorithms for the exact solution have cubic complexity in the number of datapoints, large datasets require to resort to approximations. In this work, the computation of the least-squares prediction is itself treated as a probabilistic inference problem. We propose a structured Gaussian regression model on the kernel function that uses projections of the kernel matrix to obtain a low-rank approximation of the kernel and the matrix. A central result is an enhanced way to use the method of conjugate gradients for the specific setting of least-squares regression as encountered in machine learning.

We present new excess risk bounds for general unbounded loss functions including log loss and squared loss, where the distribution of the losses may be heavy-tailed. The bounds hold for general estimators, but they are optimized when applied to $eta$-generalized Bayesian, MDL, and empirical risk minimization estimators. In the case of log loss, the bounds imply convergence rates for generalized Bayesian inference under misspecification in terms of a generalization of the Hellinger metric as long as the learning rate $eta$ is set correctly. For general loss functions, our bounds rely on two separate conditions: the $v$-GRIP (generalized reversed information projection) conditions, which control the lower tail of the excess loss; and the newly introduced witness condition, which controls the upper tail. The parameter $v$ in the $v$-GRIP conditions determines the achievable rate and is akin to the exponent in the Tsybakov margin condition and the Bernstein condition for bounded losses, which the $v$-GRIP conditions generalize; favorable $v$ in combination with small model complexity leads to $ ilde{O}(1/n)$ rates. The witness condition allows us to connect the excess risk to an 'annealed' version thereof, by which we generalize several previous results connecting Hellinger and Rényi divergence to KL divergence.

This work is concerned with the non-negative rank-1 robust principal component analysis (RPCA), where the goal is to recover the dominant non-negative principal components of a data matrix precisely, where a number of measurements could be grossly corrupted with sparse and arbitrary large noise. Most of the known techniques for solving the RPCA rely on convex relaxation methods by lifting the problem to a higher dimension, which significantly increase the number of variables. As an alternative, the well-known Burer-Monteiro approach can be used to cast the RPCA as a non-convex and non-smooth $ell_1$ optimization problem with a significantly smaller number of variables. In this work, we show that the low-dimensional formulation of the symmetric and asymmetric positive rank-1 RPCA based on the Burer-Monteiro approach has benign landscape, i.e., 1) it does not have any spurious local solution, 2) has a unique global solution, and 3) its unique global solution coincides with the true components. An implication of this result is that simple local search algorithms are guaranteed to achieve a zero global optimality gap when directly applied to the low-dimensional formulation. Furthermore, we provide strong deterministic and probabilistic guarantees for the exact recovery of the true principal components. In particular, it is shown that a constant fraction of the measurements could be grossly corrupted and yet they would not create any spurious local solution.

An important problem in the field of Topological Data Analysis is defining topological summaries which can be combined with traditional data analytic tools. In recent work Bubenik introduced the persistence landscape, a stable representation of persistence diagrams amenable to statistical analysis and machine learning tools. In this paper we generalise the persistence landscape to multiparameter persistence modules providing a stable representation of the rank invariant. We show that multiparameter landscapes are stable with respect to the interleaving distance and persistence weighted Wasserstein distance, and that the collection of multiparameter landscapes faithfully represents the rank invariant. Finally we provide example calculations and statistical tests to demonstrate a range of potential applications and how one can interpret the landscapes associated to a multiparameter module.

Derivatives play an important role in bandwidth selection methods (e.g., plug-ins), data analysis and bias-corrected confidence intervals. Therefore, obtaining accurate derivative information is crucial. Although many derivative estimation methods exist, the majority require a fixed design assumption. In this paper, we propose an effective and fully data-driven framework to estimate the first and second order derivative in random design. We establish the asymptotic properties of the proposed derivative estimator, and also propose a fast selection method for the tuning parameters. The performance and flexibility of the method is illustrated via an extensive simulation study.

In many areas, practitioners need to analyze large data sets that challenge conventional single-machine computing. To scale up data analysis, distributed and parallel computing approaches are increasingly needed. Here we study a fundamental and highly important problem in this area: How to do ridge regression in a distributed computing environment? Ridge regression is an extremely popular method for supervised learning, and has several optimality properties, thus it is important to study. We study one-shot methods that construct weighted combinations of ridge regression estimators computed on each machine. By analyzing the mean squared error in a high-dimensional random-effects model where each predictor has a small effect, we discover several new phenomena. Infinite-worker limit: The distributed estimator works well for very large numbers of machines, a phenomenon we call 'infinite-worker limit'. Optimal weights: The optimal weights for combining local estimators sum to more than unity, due to the downward bias of ridge. Thus, all averaging methods are suboptimal. We also propose a new Weighted ONe-shot DistributEd Ridge regression algorithm (WONDER). We test WONDER in simulation studies and using the Million Song Dataset as an example. There it can save at least 100x in computation time, while nearly preserving test accuracy.

We consider the problem of clustering and completing a set of tensors with missing data that are drawn from a union of low-rank tensor spaces. In the clustering problem, given a partially sampled tensor data that is composed of a number of subtensors, each chosen from one of a certain number of unknown tensor spaces, we need to group the subtensors that belong to the same tensor space. We provide a geometrical analysis on the sampling pattern and subsequently derive the sampling rate that guarantees the correct clustering under some assumptions with high probability. Moreover, we investigate the fundamental conditions for finite/unique completability for the union of tensor spaces completion problem. Both deterministic and probabilistic conditions on the sampling pattern to ensure finite/unique completability are obtained. For both the clustering and completion problems, our tensor analysis provides significantly better bound than the bound given by the matrix analysis applied to any unfolding of the tensor data.

Graphs arise naturally in many real-world applications including social networks, recommender systems, ontologies, biology, and computational finance. Traditionally, machine learning models for graphs have been mostly designed for static graphs. However, many applications involve evolving graphs. This introduces important challenges for learning and inference since nodes, attributes, and edges change over time. In this survey, we review the recent advances in representation learning for dynamic graphs, including dynamic knowledge graphs. We describe existing models from an encoder-decoder perspective, categorize these encoders and decoders based on the techniques they employ, and analyze the approaches in each category. We also review several prominent applications and widely used datasets and highlight directions for future research.

We consider the problem of estimating the latent structure of a social network based on the observed information diffusion events, or cascades, where the observations for a given cascade consist of only the timestamps of infection for infected nodes but not the source of the infection. Most of the existing work on this problem has focused on estimating a diffusion matrix without any structural assumptions on it. In this paper, we propose a novel model based on the intuition that an information is more likely to propagate among two nodes if they are interested in similar topics which are also prominent in the information content. In particular, our model endows each node with an influence vector (which measures how authoritative the node is on each topic) and a receptivity vector (which measures how susceptible the node is for each topic). We show how this node-topic structure can be estimated from the observed cascades, and prove the consistency of the estimator. Experiments on synthetic and real data demonstrate the improved performance and better interpretability of our model compared to existing state-of-the-art methods.

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.

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.

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.

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.

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 $mu_{S}$.

Andreas Anastasiou, Robert E. Gaunt.

Source: Brazilian Journal of Probability and Statistics, Volume 34, Number 1, 136--149.

Abstract:
We use the delta method and Stein’s method to derive, under regularity conditions, explicit upper bounds for the distributional distance between the distribution of the maximum likelihood estimator (MLE) of a $d$-dimensional parameter and its asymptotic multivariate normal distribution. Our bounds apply in situations in which the MLE can be written as a function of a sum of i.i.d. $t$-dimensional random vectors. We apply our general bound to establish a bound for the multivariate normal approximation of the MLE of the normal distribution with unknown mean and variance.

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.

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.

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.

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