Showing papers in "Brazilian Journal of Probability and Statistics in 2015"
TL;DR: The theory of regularity structures as discussed by the authors unifies the theory of (controlled) rough paths with the usual theory of Taylor expansions and allows to treat situations where the underlying space is multidimensional.
Abstract: We give a short introduction to the main concepts of the general theory of regularity structures. This theory unifies the theory of (controlled) rough paths with the usual theory of Taylor expansions and allows to treat situations where the underlying space is multidimensional.
75 citations
TL;DR: In this paper, a new family of distributions by compounding the Burr XII and power series distributions is introduced, which provides better fits to real data than other important lifetime models available in the literature.
Abstract: Generalizing lifetime distributions is always precious for applied statisticians In this paper, we introduce a new family of distributions by compounding the Burr XII and power series distributions The compounding procedure follows the key idea by Adamidis and Loukas ( Statist Probab Lett 39 (1998) 35–42) or, more generally, by Chahkandi and Ganjali ( Comput Statist Data Anal 53 (2009) 4433–4440) and Morais and Barreto-Souza ( Comput Statist Data Anal 55 (2011) 1410–1425) The proposed family includes as a basic exemplar the Burr XII distribution We provide some mathematical properties including moments, quantile and generating functions, order statistics and their moments, Kullback–Leibler divergence and Shannon entropy The estimation of the model parameters is performed by maximum likelihood and the inference under large sample Two special models of the new family are investigated in details We illustrate the potential of the new family by means of two applications to real data It provides better fits to these data than other important lifetime models available in the literature
66 citations
TL;DR: In this article, the authors focus on phase models, proposed and repeatedly considered by Y. Kuramoto and coauthors, and show that phase models can be reduced to an equilibrium statistical mechanics model with a very special choice of the parameters.
Abstract: Large families of noisy interacting units (cells, individuals, components in a circuit, …) exhibiting synchronization often exhibit oscillatory behaviors too. This is a well established empirical observation that has attracted a remarkable amount of attention, notably in life sciences, because of the central role played by internally generated rhythms. A certain number of elementary models that seem to capture the essence, or at least some essential features, of the phenomenon have been set forth, but the mathematical analysis is in any case very challenging and often out of reach. We focus on phase models, proposed and repeatedly considered by Y. Kuramoto and coauthors, and on the mathematical results that can be established. In spite of the fact that noise plays a crucial role, and in fact these models in abstract terms are just a special class of diffusions in high dimensional spaces, the core of the analysis is at the level of the PDE that provides an accurate description of the limit of a very large number of units in interaction. We will stress how the fundamental difficulty in dealing with these models is in their non-equilibrium character and the results we present for phase models are crucially related to the fact that, with a very special choice of the parameters, they reduce to an equilibrium statistical mechanics model.
35 citations
TL;DR: In this article, a first order non-negative integer valued autoregressive process with power series innovations based on the binomial thinning is introduced, and the main properties of the model are derived, such as mean, variance and the autocorrelation function.
Abstract: In this paper, we introduce a first order non-negative integer valued autoregressive process with power series innovations based on the binomial thinning. This new model contains, as particular cases, several models such as the Poisson INAR(1) model (Al-Osh and Alzaid (J. Time Series Anal. 8 (1987) 261–275)), the geometric INAR(1) model (Jazi, Jones and Lai (J. Iran. Stat. Soc. (JIRSS) 11 (2012) 173–190)) and many others. The main properties of the model are derived, such as mean, variance and the autocorrelation function. Yule–Walker, conditional least squares and conditional maximum likelihood estimators of the model parameters are derived. An extensive Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples. Special sub-models are studied in some detail. Applications to two real data sets are given to show the flexibility and potentiality of the new model.
27 citations
TL;DR: In this paper, a bivariate Gaussian copula is used to obtain the bivariate sinh-normal distribution, which is a symmetric distribution with three parameters and has seven parameters.
Abstract: Sinh-normal distribution is a symmetric distribution with three parameters In this paper, we introduce bivariate sinh-normal distribution, which has seven parameters Due to presence of seven parameters it is a very flexible distribution Different properties of this new distribution has been established The model can be obtained as a bivariate Gaussian copula also Therefore, using the Gaussian copula property, several properties of this proposed distribution can be obtained Maximum likelihood estimators cannot be obtained in closed forms We propose to use two step estimators based on Copula, which can be obtained in a more convenient manner One data analysis has been performed to see how the proposed model can be used in practice Finally, we consider a bivariate model which can be obtained by transforming the sinh-normal distribution and it is a generalization of the bivariate Birnbaum–Saunders distribution Several properties of the bivariate Birnbaum–Saunders distribution can be obtained as special cases of the proposed generalized bivariate Birnbaum–Saunders distribution
25 citations
TL;DR: In this paper, the existence and uniqueness of a non-equilibrium steady state (NESS) was proved and convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wennberg (GTW).
Abstract: We investigate a kinetic model of a system in contact with several thermal reservoirs at different temperatures $T_\alpha$. Our system is a spatially uniform dilute gas whose internal dynamics is described by the non-linear Boltzmann equation with Maxwellian collisions. Similarly, the interaction with reservoir $\alpha$ is represented by a Markovian process that has the Maxwellian $M_{T_\alpha}$ as its stationary state. We prove existence and uniqueness of a non-equilibrium steady state (NESS) and show exponential convergence to this NESS in a metric on probability measures introduced into the study of Maxwellian collisions by Gabetta, Toscani and Wennberg (GTW). This shows that the GTW distance between the current velocity distribution to the steady-state velocity distribution is a Lyapunov functional for the system. We also derive expressions for the entropy production in the system plus the reservoirs which is always positive.
21 citations
TL;DR: In this article, a statistical methodology based on matrix-variate Birnbaum-Saunders (BS) distributions is proposed to model diverse phenomena from the real-world using random matrices.
Abstract: Diverse phenomena from the real-world can be modeled using random matrices, allowing matrix-variate distributions to be considered. The normal distribution is often employed in this modeling, but usually the mentioned random matrices do not follow such a distribution. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum–Saunders (BS) distribution. We propose a statistical methodology based on matrix-variate BS distributions. This methodology is implemented in the statistical software R. A simulation study is conducted to evaluate its performance. Finally, an application with real-world matrix-variate data is carried out to illustrate its potentiality and suitability.
19 citations
TL;DR: In this paper, the authors investigated the consistency of the mean and variance of the sub-fractional Brownian motion with drift at discrete observation and obtained the central limit theorem and the Berry-Esseen bounds for these estimators.
Abstract: In this paper, we investigate the $L^{2}$-consistency and the strong consistency of the maximum likelihood estimators (MLE) of the mean and variance of the sub-fractional Brownian motion with drift at discrete observation. By combining the Stein’s method with Malliavin calculus, we obtain the central limit theorem and the Berry–Esseen bounds for these estimators.
19 citations
TL;DR: In this paper, it was shown that the polyspectra can be given as the coefficients of the orthogonal expansion of cumulants of the field in terms of irreducible tensor products of spherical harmonics.
Abstract: Cosmic microwave background (CMB) Anisotropies is a subject of intensive research in recent years, and therefore it is necessary to develop suitable theory and methods for the analysis of isotropic fields on spheres. The main object of our paper is to show that the polyspectra can be given as the coefficients of the orthogonal expansion of cumulants of the field in terms of irreducible tensor products of spherical harmonics. We obtain necessary and sufficient conditions for isotropy of a non-Gaussian field and the conditions are stated in terms of higher order spectra (polyspectra). The relation between cumulants and spectra gives a new method of estimating spectra.
16 citations
TL;DR: In this article, the problem of estimating the Renyi entropy is considered and the uniformly minimum variance unbiased estimator (UMVUE) is derived, and sufficient conditions for improvement over affine and scale equivariant estimators are obtained.
Abstract: Suppose independent random samples are drawn from $k$ shifted exponential populations with a common location but unequal scale parameters. The problem of estimating the Renyi entropy is considered. The uniformly minimum variance unbiased estimator (UMVUE) is derived. Sufficient conditions for improvement over affine and scale equivariant estimators are obtained. As a consequence, improved estimators over the UMVUE and the maximum likelihood estimator (MLE) are obtained. Further, for the case $k=1$, an estimator that dominates the best affine equivariant estimator is derived. Cases when the location parameter is constrained are also investigated in detail.
14 citations
TL;DR: In this article, the authors derived maximum likelihood estimators, uniformly minimum variance unbiased estimators and three other estimators of the probability density function and the cumulative distribution function of the exponentiated Weibull distribution and compare their performances through numerical simulations.
Abstract: Exponentiated Weibull distribution, introduced as an extension of the Weibull distribution, is characterized by bathtub shaped, unimodal failure rates besides a broader class of monotone failure rates. In this paper, we derive maximum likelihood estimators (MLEs), uniformly minimum variance unbiased estimators and three other estimators of the probability density function and the cumulative distribution function of the exponentiated Weibull distribution and compare their performances through numerical simulations. Simulation studies show that the MLE is more efficient than the others. Analysis of a real data set is presented for illustrative purposes.
TL;DR: In this article, the authors consider discrete-time stationary processes with long-range dependencies and give an explicit set of conditions that imply the existence of at least two distinct processes specified by the same $g$-function, and consider the role played by the smoothness of the majority rule at the origin.
Abstract: We consider discrete-time stationary processes with long-range dependencies, $X_{n}\in\{\pm1\}$, ${n\in\mathbb{Z}}$, specified by a regular attractive $g$-function, similar to those considered by Bramson and Kalikow [Israel J Math 84 (1993) 153–160] We give an explicit set of conditions that imply the existence of at least two distinct processes specified by the same $g$-function, and consider a few examples that emphasize the role played by the smoothness of the majority rule at the origin
TL;DR: In this article, the authors considered the GINAR(1) process with exponential margin and derived large sample distributions of the conditional least squares estimates of the model parameters and performed a numerical study to illustrate the theoretical results.
Abstract: In the context of an over-dispersed count time series data on disease incidences, we consider the Geometric integer-valued autoregressive process of order 1 or GINAR(1), which was first introduced by McKenzie (Adv. Appl. Probab. 18 (1986) 679–705) as an analogue of continuous AR(1) process with exponential margin (Adv. Appl. Probab. 12 (1980) 727–745) on the positive support ($\mathbb{R}^{+}$). A strong enthusiasm still persists as it is apparent from Ristic et al. (J. Stat. Plann. Inf. 139 (2009) 2218–2226). Coherent forecasting of Poisson INAR(1) process due to Al-Osh and Alzaid (J. Time Ser. Anal. 8 (1987) 261–275) was studied by Freeland and McCabe (Int. J. Forecast. 20 (2004) 427–434). Here, we study the $h$-step ahead forecasting distribution corresponding to GINAR(1) process in details using probability generating function. Large sample distributions of the conditional least squares estimates of the model parameters are derived. Some numerical study is performed to illustrate the theoretical results.
TL;DR: In this article, the authors extend the generalized time-dependent logistic model by adding a frailty term in the modeling and use the Laplace transform to find the survival function unconditional on the individual frailty.
Abstract: Survival models with univariate frailty may be used when there is no information on covariates that are important to explain the failure time. The lack of information may be with respect to covariates that were not observed or even covariates which for some reason we can not measure, for instance, environmental or genetic factors. In this paper, we extend the generalized time-dependent logistic model proposed by Mackenzie (The Statistician 45 (1996) 21–34), by including a frailty term in the modeling. The proposed methodology uses the Laplace transform to find the survival function unconditional on the individual frailty. A simulation study examines the bias, the mean squared errors and the coverage probabilities. Estimation is based on maximum likelihood. A real example on lung cancer illustrates the applicability of the methodology, which is compared to the modeling without frailty via selection criteria to determine which model best fits the data.
TL;DR: In this article, the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity ε(n) is investigated.
Abstract: We consider the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity $\epsilon^{\frac 12}$, and we investigate the effect of the noise, as $\epsilon \to 0$, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given $\gamma< \frac 23$, with probability going to one as $\epsilon \to 0$, the solution remains close to a front for times of the order of $\epsilon^{-\gamma}$, and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order $\frac 14$ and non Markovian, related to a fractional Brownian motion and for which a couple of representations are given.
TL;DR: In this article, the authors proposed nonparametric estimators for the residual entropy function based on censored data, and compared the performance of the estimators using the mean-squared error.
Abstract: The residual entropy function introduced by Ebrahimi [Sankhyā A 58 (1996) 48–56], is viewed as a dynamic measure of uncertainty. This measure finds applications in modeling and analysis of life time data. In the present work, we propose nonparametric estimators for the residual entropy function based on censored data. Asymptotic properties of the estimator are established under suitable regularity conditions. Monte Carlo simulation studies are carried out to compare the performance of the estimators using the mean-squared error. The methods are illustrated using two real data sets.
TL;DR: In the last decade, the theory of large deviations has become a main tool in statistical mechanics especially in the study of non-equilibrium as discussed by the authors, and it has been a source of inspiration in recent developments of nonequilibrium statistical mechanics.
Abstract: In the last decade, the theory of large deviations has become a main tool in statistical mechanics especially in the study of non-equilibrium. In a rational reconstruction of the story, one must recognize the ideal connection and debt of some recent work, to discussions taking place at the beginning of the twentieth century. The famous equation $S=k\ln W$ usually attributed to Boltzmann, actually written in this final form by Planck on his route to the quantum hypothesis, was interpreted by Einstein as a large deviation formula. This interpretation, on which he based his theory of thermodynamic equilibrium fluctuations, has been a source of inspiration in recent developments of non-equilibrium statistical mechanics. In this paper, we briefly illustrate this aspect.
TL;DR: In this article, a Bayesian Factor Model (BFM) is proposed to identify coherent patterns in gene expression data sets. But, the BFM does not address the problem of cross-hybridization, where a given probe is not measuring its targeted gene, but rather a different gene with a similar region.
Abstract: A common problem in the analysis of gene expression microarray data is the identification of groups of features that are coherently expressed. For example, one often wishes to know whether a group of genes, clustered because of correlation in one data set, are still highly co-expressed in another data set. Alternatively, for some expression array platforms there are many, relatively short probes for each gene of interest. In this case, it is possible that a given probe is not measuring its targeted gene, but rather a different gene with a similar region (called cross-hybridization). Accurate detection of the collection of probe sets (groups of probes targeting the same gene) which demonstrate highly coherent expression patterns is the best approach to the identification of which genes are present in the sample. We develop a Bayesian Factor Model (BFM) to address the general problem of detection of coherent patterns in gene expression data sets. We compare our method to “state of the art” methods for the identification of expressed genes in both synthetic and real data sets, and the results indicate that the BFM outperforms the other procedures for detecting transcripts. We also demonstrate the use of factor analysis to identify the presence/absence status of gene modules (groups of coherently expressed genes). Variation in the number of copies of regions of the genome is a well known and important feature of most cancers. We examine a group of genes, representative of Copy Number Alteration (CNA) in breast cancer, then identify the presence/absence of CNA in this region of the genome for other cancers. Coherent patterns can also be evaluated in high-throughput sequencing data, a novel technology to measure gene expression. We analyze this type of data via factor model and examine the detection calls in terms of read mapping uncertainty.
TL;DR: In this article, the authors derived an upper bound on the Wasserstein distance between the distribution of a random recursive tree and a standard normal distribution and obtained a weak version of an Erdos-Renyi type law and a large deviation principle for the number of leaves.
Abstract: This paper studies the asymptotic behavior of the number of leaves $L_{n}$ in a random recursive tree $T_{n}$ with $n$ nodes. By utilizing the size-bias method, we derive an upper bound on the Wasserstein distance between the distribution of $L_{n}$ and a standard normal distribution. Furthermore, we obtain a weak version of an Erdos–Renyi type law and a large deviation principle for $L_{n}$.
TL;DR: In this paper, the authors consider a finite number of particles that move in Z as independent random walks and prove that in the hydrodynamic limit the evolution is described by a nonlinear system of two PDEs with free boundaries.
Abstract: We consider a finite number of particles that move in $\mathbb{Z}$ as independent random walks. The particles are of two species that we call $a$ and $b$. The rightmost $a$-particle becomes a $b$-particle at constant rate, while the leftmost $b$-particle becomes $a$-particle at the same rate, independently. We prove that in the hydrodynamic limit the evolution is described by a nonlinear system of two PDE’s with free boundaries.
TL;DR: In this paper, the authors obtain macroscopic adiabatic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment, and prove that the relations between the limit work, internal energy and thermodynamic entropy agree with the first and second principle of thermodynamics.
Abstract: We obtain macroscopic adiabatic thermodynamic transformations by space-time scalings of a microscopic Hamiltonian dynamics subject to random collisions with the environment. The microscopic dynamics is given by a chain of oscillators subject to a varying tension (external force) and to collisions with external independent particles of ''infinite mass''. The effect of each collision is to change the sign of the velocity without changing the modulus. This way the energy is conserved by the resulting dynamics. After a diffusive space-time scaling and cross-graining, the profiles of volume and energy converge to the solution of a deterministic diffusive system of equations with boundary conditions given by the applied tension. This defines an irreversible thermodynamic transformation from an initial equilibrium to a new equilibrium given by the final tension applied. Quasi-static reversible adiabatic transformations are then obtained by a further time scaling. Then we prove that the relations between the limit work, internal energy and thermodynamic entropy agree with the first and second principle of thermodynamics.
TL;DR: In this paper, necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measures are discussed, where the authors consider the case where the process is asymmetric.
Abstract: We discuss necessary and sufficient conditions for the convergence of disordered asymmetric zero-range process to the critical invariant measures.
TL;DR: In this paper, a sub-Gaussian lower bound for the transition kernel of the one-dimensional, symmetric Bouchaud trap model is established, which provides a positive answer to the behavior predicted by Bertin and Bouchau in 2013.
Abstract: We establish a sub-Gaussian lower bound for the transition kernel of the one-dimensional, symmetric Bouchaud trap model, which provides a positive answer to the behavior predicted by Bertin and Bouchaud in (Phys. Rev. E (3) 67 (2013) 026128). The proof rests on the Ray–Knight description of the local time of a one-dimensional Brownian motion. Using the same ideas, we also prove the corresponding result for the FIN singular diffusion.
TL;DR: In this article, the problem of identifying pairs of interacting sites from a finite sample of independent realizations of the Ising model is addressed, and an upper bound for the probability of misidentification of the pairs of interaction sites in this finite set is derived.
Abstract: This paper address the problem of identifying pairs of interacting sites from a finite sample of independent realizations of the Ising model. We consider Ising models in a infinite countable set of sites under Dobrushin uniqueness condition. The observed sample contains only the values assigned by the Ising model to a finite set of sites. Our main result is an upperbound for the probability of misidentification of the pairs of interacting sites in this finite set.
TL;DR: In this article, the authors proved an almost sure central limit theorem for the exceedance point processes of a stationary sequence which satisfy some long range dependence conditions, and extended the results to the vector of point processes for strong mixing random sequences.
Abstract: In this paper, we proved an almost sure central limit theo- rem for the exceedance point processes of a stationary sequence which satisfy some long range dependence conditions. As a by-product, we ob- tained the almost sure central limit theorem for the order statistics of the stationary sequence. The obtained results are also extended to the vector of point processes for some strong mixing random sequences.
TL;DR: In this article, the authors use Holley inequality to obtain sufficient conditions under which the following hold: the trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory starting from any probability on the countable set when both are conditioned to nonabsorption until a certain time.
Abstract: Let ${S}$ be a countable set provided with a partial order and a minimal element. Consider a Markov chain on $S\cup\{0\}$ absorbed at $0$ with a quasi-stationary distribution. We use Holley inequality to obtain sufficient conditions under which the following hold. The trajectory of the chain starting from the minimal state is stochastically dominated by the trajectory of the chain starting from any probability on ${S}$, when both are conditioned to nonabsorption until a certain time. Moreover, the Yaglom limit corresponding to this deterministic initial condition is the unique minimal quasi-stationary distribution in the sense of stochastic order. As an application, we provide new proofs to classical results in the field.
TL;DR: These are the notes of my lectures at EBP, August 2013 as mentioned in this paper, which are based on a course on atomistic and continuous descriptions of matter which I gave a few years earlier in Sperlonga, Italy.
Abstract: These are the notes of my lectures at EBP, August 2013. I have added some proofs which being of more technical nature have been omitted in the talks. The notes and the lectures are based on a course on atomistic and continuous descriptions of matter which I gave a few years earlier in Sperlonga, Italy. In the present notes, I have tried to underline the more probabilistic aspects of the theory. I am afraid I have not been able to reproduce in the written notes the very lively atmosphere of the talks. Many old friends of mine were attending the lectures and helped me a lot with questions, comments and criticism, it was a pleasure for me to speak at EBP and I hope also the audience enjoined all that. In particular, I want to renew my deepest thanks to Stefan Luckhaus who is undoubtedly the best help for a speaker to have in the audience. These notes are divided into four chapters, like in the lectures I have mostly avoided proofs trying to give qualitatively the main ideas of the theory. The only exception is in the second lecture of these notes where I have given more details on the proof of phase transitions in the canonical Ising model. This partly for completeness and partly because I have been asked by several people for details after the talk. I am not very good with bibliography so I just quoted some of the papers I am most familiar with, the reader will forgive me, I hope, for the many omissions. I conclude these preface by renewing my warmest thanks to the people who attended EBP and in particular to Maria Eulalia Vares for inviting me at EBP, for the nice words when she introduced my lectures but especially for the very long friendship (in the past and hopefully in the future).
TL;DR: In this paper, the density function of the Harris-G class can be expressed as a linear combination of density functions of the exponentiated baseline distribution, and general formulas for moments (raw, centered, incomplete and factorial), quantile function, generating functions and entropies are provided.
Abstract: We investigate some properties of the Harris-G class of distributions (Sankhya B 73 (2011) 70–80). We demonstrate that the density function of the Harris-G class can be expressed as a linear combination of density functions of the exponentiated baseline distribution. We provide general formulas for moments (raw, centered, incomplete and factorial), quantile function, generating functions and entropies. Two numerical examples are presented to demonstrate the potentiality of the models in this class. The first one applies the Harris–Burr XII distribution to model bimodal data. The second example uses the Harris-exponential distribution to model SAR image data. The results of the fitted models look promising.