Brazilian Journal of Probability and Statistics Articles (Project Euclid)
http://projecteuclid.org/euclid.bjps
The latest articles from Brazilian Journal of Probability and Statistics on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTThu, 31 Mar 2011 09:13 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
http://projecteuclid.org/
An estimation method for latent traits and population parameters in Nominal Response Model
http://projecteuclid.org/euclid.bjps/1280754493
<strong>Caio L. N. Azevedo</strong>, <strong>Dalton F. Andrade</strong><p><strong>Source: </strong>Braz. J. Probab. Stat., Volume 24, Number 3, 415--433.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1280754493_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTImproved estimation in a general multivariate elliptical modelhttps://projecteuclid.org/euclid.bjps/1520046134<strong>Tatiane F. N. Melo</strong>, <strong>Silvia L. P. Ferrari</strong>, <strong>Alexandre G. Patriota</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 44--68.</p><p><strong>Abstract:</strong><br/>
The problem of reducing the bias of maximum likelihood estimator in a general multivariate elliptical regression model is considered. The model is very flexible and allows the mean vector and the dispersion matrix to have parameters in common. Many frequently used models are special cases of this general formulation, namely: errors-in-variables models, nonlinear mixed-effects models, heteroscedastic nonlinear models, among others. In any of these models, the vector of the errors may have any multivariate elliptical distribution. We obtain the second-order bias of the maximum likelihood estimator, a bias-corrected estimator, and a bias-reduced estimator. Simulation results indicate the effectiveness of the bias correction and bias reduction schemes.
</p>projecteuclid.org/euclid.bjps/1520046134_20180302220218Fri, 02 Mar 2018 22:02 ESTImproved inference for the generalized Pareto distributionhttps://projecteuclid.org/euclid.bjps/1520046135<strong>Juliana F. Pires</strong>, <strong>Audrey H. M. A. Cysneiros</strong>, <strong>Francisco Cribari-Neto</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 69--85.</p><p><strong>Abstract:</strong><br/>
The generalized Pareto distribution is commonly used to model exceedances over a threshold. In this paper, we obtain adjustments to the generalized Pareto profile likelihood function using the likelihood function modifications proposed by Barndorff-Nielsen ( Biometrika 70 (1983) 343–365), Cox and Reid ( J. R. Stat. Soc. Ser. B. Stat. Methodol. 55 (1993) 467–471), Fraser and Reid ( Utilitas Mathematica 47 (1995) 33–53), Fraser, Reid and Wu ( Biometrika 86 (1999) 249–264) and Severini ( Biometrika 86 (1999) 235–247). We consider inference on the generalized Pareto distribution shape parameter, the scale parameter being a nuisance parameter. Bootstrap-based testing inference is also considered. Monte Carlo simulation results on the finite sample performances of the usual profile maximum likelihood estimator and profile likelihood ratio test and also their modified versions is presented and discussed. The numerical evidence favors the modified profile maximum likelihood estimators and tests we propose. Finally, we consider two real datasets as illustrations.
</p>projecteuclid.org/euclid.bjps/1520046135_20180302220218Fri, 02 Mar 2018 22:02 ESTNonlinear measurement errors models subject to partial linear additive distortionhttps://projecteuclid.org/euclid.bjps/1520046136<strong>Jun Zhang</strong>, <strong>Nanguang Zhou</strong>, <strong>Qian Chen</strong>, <strong>Tianyue Chu</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 86--116.</p><p><strong>Abstract:</strong><br/>
We study nonlinear regression models when the response and predictors are unobservable and distorted in a multiplicative fashion by partial linear additive models (PLAM) of some observed confounding variables. After approximating the additive nonparametric components in the PLAM via polynomial splines and calibrating the unobserved response and unobserved predictors, we develop a semi-parametric profile nonlinear least squares procedure to estimate the parameters of interest. The resulting estimators are shown to be asymptotically normal. To construct confidence intervals for the parameters of interest, an empirical likelihood-based statistic is proposed to improve the accuracy of the associated normal approximation. We also show that the empirical likelihood statistic is asymptotically chi-squared. Moreover, a test procedure based on the empirical process is proposed to check whether the parametric regression model is adequate or not. A wild bootstrap procedure is proposed to compute $p$-values. Simulation studies are conducted to examine the performance of the estimation and testing procedures. The methods are applied to re-analyze real data from a diabetes study for an illustration.
</p>projecteuclid.org/euclid.bjps/1520046136_20180302220218Fri, 02 Mar 2018 22:02 ESTOn the exit time from an orthant for badly oriented random walkshttps://projecteuclid.org/euclid.bjps/1520046137<strong>Rodolphe Garbit</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 117--146.</p><p><strong>Abstract:</strong><br/>
It was recently proved that the exponential decreasing rate of the probability that a random walk stays in a $d$-dimensional orthant is given by the minimum on this orthant of the Laplace transform of the random walk increments, provided that this minimum exists. In other cases, the random walk is “badly oriented” and the exponential rate may depend on the starting point $x$. We show here that this rate is nevertheless asymptotically equal to the infimum of the Laplace transform, as some selected coordinates of $x$ tend to infinity.
</p>projecteuclid.org/euclid.bjps/1520046137_20180302220218Fri, 02 Mar 2018 22:02 ESTNoise-indicator nonnegative integer-valued autoregressive time series of the first orderhttps://projecteuclid.org/euclid.bjps/1520046138<strong>Vladica Stojanović</strong>, <strong>Dragan Randjelović</strong>, <strong>Kristijan Kuk</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 147--171.</p><p><strong>Abstract:</strong><br/>
This paper presents a modification and, at the same time, a generalization of the linear first order nonnegative integer-valued autoregressive processes, well-known as INAR(1) processes. By using the so-called Noise-Indicator, a nonlinear model with the threshold regime and with more complex structure than the appropriate linear models was obtained. The new model, named NIINAR(1) process, has been investigated in terms of the most general, the power series distribution of its innovations. Basic stochastic properties of the NIINAR(1) model (e.g., correlation structure, over-dispersion conditions and distributional properties) are given. Also, besides of some standard parameters estimators, a novel estimation techniques, together with the asymptotic properties of the obtained estimates is described. At last, a Monte Carlo study of this process is also given, as well as its application in the analysis of dynamics of two empirical dataset.
</p>projecteuclid.org/euclid.bjps/1520046138_20180302220218Fri, 02 Mar 2018 22:02 ESTHölderian weak invariance principle under the Maxwell and Woodroofe conditionhttps://projecteuclid.org/euclid.bjps/1520046139<strong>Davide Giraudo</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 172--187.</p><p><strong>Abstract:</strong><br/>
We investigate the weak invariance principle in Hölder spaces under some reinforcement of the Maxwell and Woodroofe condition. Optimality of the obtained condition is established.
</p>projecteuclid.org/euclid.bjps/1520046139_20180302220218Fri, 02 Mar 2018 22:02 ESTAbrupt convergence for a family of Ornstein–Uhlenbeck processeshttps://projecteuclid.org/euclid.bjps/1520046140<strong>Gerardo Barrera</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 188--199.</p><p><strong>Abstract:</strong><br/>
We consider a family of Ornstein–Uhlenbeck processes. Under some suitable assumptions on the behaviour of the drift and diffusion coefficients, we prove profile cut-off phenomenon with respect to the total variation distance in the sense of the definition given by Barrera and Ycart [ ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 445–458]. We compute explicitly the cut-off time, the window time, and the profile function. Moreover, we prove that the average process satisfies a profile cut-off phenomenon with respect to the total variation distance. Also, a sample of $N$ Ornstein–Uhlenbeck processes has a window cut-off with respect to the total variation distance in the sense of the definition given by Barrera and Ycart [ ALEA Lat. Am. J. Probab. Math. Stat. 11 (2014) 445–458]. The cut-off time and the cut-off window for the average process and for the sampling process are the same.
</p>projecteuclid.org/euclid.bjps/1520046140_20180302220218Fri, 02 Mar 2018 22:02 ESTA Skellam GARCH modelhttps://projecteuclid.org/euclid.bjps/1520046141<strong>Ghadah A. Alomani</strong>, <strong>Abdulhamid A. Alzaid</strong>, <strong>Maha A. Omair</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 1, 200--214.</p><p><strong>Abstract:</strong><br/>
This paper considers the modeling of nonstationary integer valued time series with conditional heteroskedasticity using Skellam distribution. Two approaches of estimation of the model’s parameters are treated and discussed. The obtained results are verified through some numerical simulation. In addition, the proposed model is applied to real time series.
</p>projecteuclid.org/euclid.bjps/1520046141_20180302220218Fri, 02 Mar 2018 22:02 ESTTruncated sequential Monte Carlo test with exact powerhttps://projecteuclid.org/euclid.bjps/1523952013<strong>Ivair Silva</strong>, <strong>Renato Assunção</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 215--238.</p><p><strong>Abstract:</strong><br/>
Monte Carlo hypothesis testing is extensively used for statistical inference. Surprisingly, despite the many theoretical advances in the field, statistical power performance of Monte Carlo tests remains an open question. Because the last assertion may sound questionable for some, the first goal in this paper is to show that the power performance of truncated Monte Carlo tests is still an unsolved question. The second goal here is to present a solution for this issue, that is, we introduce a truncated sequential Monte Carlo procedure with statistical power arbitrarily close to the power of the theoretical exact test. The most significant contribution of this work is the validity of our method for the general case of any test statistic.
</p>projecteuclid.org/euclid.bjps/1523952013_20180417040017Tue, 17 Apr 2018 04:00 EDTBayesian analysis of multiple-inflation Poisson models and its application to infection datahttps://projecteuclid.org/euclid.bjps/1523952014<strong>Duchwan Ryu</strong>, <strong>Devrim Bilgili</strong>, <strong>Önder Ergönül</strong>, <strong>Nader Ebrahimi</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 239--261.</p><p><strong>Abstract:</strong><br/>
In this article we propose a multiple-inflation Poisson regression to model count response data containing excessive frequencies at more than one non-negative integer values. To handle multiple excessive count responses, we generalize the zero-inflated Poisson regression by replacing its binary regression with the multinomial regression, while Su et al. [ Statist. Sinica 23 (2013) 1071–1090] proposed a multiple-inflation Poisson model for consecutive count responses with excessive frequencies. We give several properties of our proposed model, and do statistical inference under the fully Bayesian framework. We perform simulation studies and also analyze the data related to the number of infections collected in five major hospitals in Turkey, using our methodology.
</p>projecteuclid.org/euclid.bjps/1523952014_20180417040017Tue, 17 Apr 2018 04:00 EDTPoisson–Lindley INAR(1) model with applicationshttps://projecteuclid.org/euclid.bjps/1523952015<strong>M. Mohammadpour</strong>, <strong>Hassan S. Bakouch</strong>, <strong>M. Shirozhan</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 262--280.</p><p><strong>Abstract:</strong><br/>
The paper focuses on a new stationary integer-valued autoregressive model of first order with Poisson–Lindley marginal distribution. Several statistical properties of the model are established, like spectral density function, multi-step ahead conditional measures, stationarity, ergodicity and irreducibility. We consider several methods for estimating the unknown parameters of the model and investigate properties of the estimators. The performances of these estimators are compared via simulation. The model is motivated by some applications to two real count time series data.
</p>projecteuclid.org/euclid.bjps/1523952015_20180417040017Tue, 17 Apr 2018 04:00 EDTThe exponentiated logarithmic generated family of distributions and the evaluation of the confidence intervals by percentile bootstraphttps://projecteuclid.org/euclid.bjps/1523952016<strong>Pedro Rafael Diniz Marinho</strong>, <strong>Gauss M. Cordeiro</strong>, <strong>Fernando Peña Ramírez</strong>, <strong>Morad Alizadeh</strong>, <strong>Marcelo Bourguignon</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 281--308.</p><p><strong>Abstract:</strong><br/>
We study some mathematical properties of a new generator of continuous distributions with three additional parameters, called the exponentiated logarithmic generated family, to extend the normal, Weibull, gamma and Gumbel distributions, among other well-known models. Some special models are discussed. Many properties of this family are studied, some inference procedures developed and a simulation study performed to verify the adequacy of the estimators of the model parameters. We prove empirically the potentiality of the new family by means of two real data sets. The simulation study for different samples sizes assesses the performance of the maximum likelihood estimates obtained by the Swarm Optimization method. We also evaluate the performance of single and dual bootstrap methods in constructing interval estimates for the parameters. Because of the intensive simulations, we use parallel computing on a supercomputer.
</p>projecteuclid.org/euclid.bjps/1523952016_20180417040017Tue, 17 Apr 2018 04:00 EDTOn the number of unobserved and observed categories when sampling from a multivariate hypergeometric populationhttps://projecteuclid.org/euclid.bjps/1523952017<strong>Sungsu Kim</strong>, <strong>Chong Jin Park</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 309--319.</p><p><strong>Abstract:</strong><br/>
Consider taking a random sample of size $n$ from a finite population that consists of $N$ categories with $M_{i}$ copies in the $i$th category for $i=1,\dots,N$. Each observed unit in a sample is presumed to have a probability $1-p$ ($0<p<1$) of getting lost from the sample. Let $S$ denote the number of categories not observed in the sample and $S_{j}$ denote the number of categories where $j$ samples are observed for $j=1,\dots,n$. In this paper, the probability distribution and factorial moments of $S$ and $S_{j}$ are studied. A matrix inversion algorithm is used in order to facilitate numerical computations in obtaining the probabilities and factorial moments. A couple of examples of the problem considered in this paper may include a filing or storage process, where objects are randomly assigned to files or storage bins, and from time to time, objects may be missing or have disappeared, species as categories in a capture-recapture problem, or DNA sequence study.
</p>projecteuclid.org/euclid.bjps/1523952017_20180417040017Tue, 17 Apr 2018 04:00 EDTMixture models applied to heterogeneous populationshttps://projecteuclid.org/euclid.bjps/1523952018<strong>Carolina V. Cavalcante</strong>, <strong>Kelly C. M. Gonçalves</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 320--345.</p><p><strong>Abstract:</strong><br/>
Mixture models provide a flexible representation of heterogeneity in a finite number of latent classes. From the Bayesian point of view, Markov Chain Monte Carlo methods provide a way to draw inferences from these models. In particular, when the number of subpopulations is considered unknown, more sophisticated methods are required to perform Bayesian analysis. The Reversible Jump Markov Chain Monte Carlo is an alternative method for computing the posterior distribution by simulation in this case. Some problems associated with the Bayesian analysis of these class of models are frequent, such as the so-called “label-switching” problem. However, as the level of heterogeneity in the population increases, these problems are expected to become less frequent and the model’s performance to improve. Thus, the aim of this work is to evaluate the normal mixture model fit using simulated data under different settings of heterogeneity and prior information about the mixture proportions. A simulation study is also presented to evaluate the model’s performance considering the number of components known and estimating it. Finally, the model is applied to a censored real dataset containing antibody levels of Cytomegalovirus in individuals.
</p>projecteuclid.org/euclid.bjps/1523952018_20180417040017Tue, 17 Apr 2018 04:00 EDTIdentifiability of structural characteristics: How relevant is it for the Bayesian approach?https://projecteuclid.org/euclid.bjps/1523952019<strong>Ernesto San Martín</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 346--373.</p><p><strong>Abstract:</strong><br/>
The role of identification in the Bayesian approach is still debatable. Since Lindley [Bayesian Statistics. A Review (1971) Philadelphia], most Bayesian statisticians pretend that unidentifiabiity causes no real difficulty in their approach. Recently, Wechsler, Izbicki and Esteves [ Amer. Statist. 67 (2013) 90–93] provide a simple example illustrating this perspective. By critically reading Wechsler, Izbicki and Esteves [ Amer. Statist. 67 (2013) 90–93], we intend to show that the Bayesian approach is far from being free of the identification problems, provided that the interest is focused on the interpretation of the parameters. It is written using a rather ancient style, the so-called Platonic dialogues. In modern times, there are beautiful examples of that, particularly in Foundations of Mathematics, where debatable subjects are discussed: let us refer Heyting [Intuitionism. An Introduction (1971) North-Holland Publishing Company], where the debate between a formalist and an intuitionist is presented as a dialogue; or Lakatos [Proofs and Refutations. The Logic of Mathematical Discovery (1976) Cambridge University Press], where the relationship between proofs and conjectures is magnificently illustrated. We hope that this style will help to understand why identifiability really matters in the Bayesian approach.
</p>projecteuclid.org/euclid.bjps/1523952019_20180417040017Tue, 17 Apr 2018 04:00 EDTNonlinear filtering with correlated Lévy noise characterized by copulashttps://projecteuclid.org/euclid.bjps/1523952020<strong>B. P. W. Fernando</strong>, <strong>E. Hausenblas</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 374--421.</p><p><strong>Abstract:</strong><br/>
The objective in stochastic filtering is to reconstruct the information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process.
Usually $X$ and $Y$ are modeled by stochastic differential equations driven by a Brownian motion or a jump (or Lévy) process. We are interested in the situation where both the state process $X$ and the observation process $Y$ are perturbed by coupled Lévy processes. More precisely, $L=(L_{1},L_{2})$ is a $2$-dimensional Lévy process in which the structure of dependence is described by a Lévy copula. We derive the associated Zakai equation for the density process and establish sufficient conditions depending on the copula and $L$ for the solvability of the corresponding solution to the Zakai equation. In particular, we give conditions of existence and uniqueness of the density process, if one is interested to estimate quantities like $\mathbb{P}(X(t)>a)$, where $a$ is a threshold.
</p>projecteuclid.org/euclid.bjps/1523952020_20180417040017Tue, 17 Apr 2018 04:00 EDTSome unified results on stochastic properties of residual lifetimes at random timeshttps://projecteuclid.org/euclid.bjps/1523952021<strong>Neeraj Misra</strong>, <strong>Sameen Naqvi</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 422--436.</p><p><strong>Abstract:</strong><br/>
The residual life of a random variable $X$ at random time $\Theta$ is defined to be a random variable $X_{\Theta}$ having the same distribution as the conditional distribution of $X-\Theta$ given $X>\Theta$ (denoted by $X_{\Theta}=(X-\Theta|X>\Theta)$). Let $(X,\Theta_{1})$ and $(Y,\Theta_{2})$ be two pairs of jointly distributed random variables, where $X$ and $\Theta_{1}$ (and, $Y$ and $\Theta_{2}$) are not necessarily independent. In this paper, we compare random variables $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ by providing sufficient conditions under which $X_{\Theta_{1}}$ and $Y_{\Theta_{2}}$ are stochastically ordered with respect to various stochastic orderings. These comparisons have been made with respect to hazard rate, likelihood ratio and mean residual life orders. We also study various ageing properties of random variable $X_{\Theta_{1}}$. By considering this generalized model, we generalize and unify several results in the literature on stochastic properties of residual lifetimes at random times. Some examples to illustrate the application of the results derived in the paper are also presented.
</p>projecteuclid.org/euclid.bjps/1523952021_20180417040017Tue, 17 Apr 2018 04:00 EDTProducts of normal, beta and gamma random variables: Stein operators and distributional theoryhttps://projecteuclid.org/euclid.bjps/1523952022<strong>Robert E. Gaunt</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 2, 437--466.</p><p><strong>Abstract:</strong><br/>
In this paper, we extend Stein’s method to products of independent beta, gamma, generalised gamma and mean zero normal random variables. In particular, we obtain Stein operators for mixed products of these distributions, which include the classical beta, gamma and normal Stein operators as special cases. These operators lead us to closed-form expressions involving the Meijer $G$-function for the probability density function and characteristic function of the mixed product of independent beta, gamma and central normal random variables.
</p>projecteuclid.org/euclid.bjps/1523952022_20180417040017Tue, 17 Apr 2018 04:00 EDTImproving mean estimation in ranked set sampling using the Rao regression-type estimatorhttps://projecteuclid.org/euclid.bjps/1528444868<strong>Elvira Pelle</strong>, <strong>Pier Francesco Perri</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 467--496.</p><p><strong>Abstract:</strong><br/>
Ranked set sampling is a statistical technique usually used for a variable of interest that may be difficult or expensive to measure, but whose units are simple to rank according to a cheap sorting criterion. In this paper, we revisit the Rao regression-type estimator in the context of the ranked set sampling. The expression of the minimum mean squared error is given and a comparative study, based on simulated and real data, is carried out to clearly show that the considered estimator outperforms some competitive estimators discussed in the recent literature.
</p>projecteuclid.org/euclid.bjps/1528444868_20180608040117Fri, 08 Jun 2018 04:01 EDTA large class of new bivariate copulas and their propertieshttps://projecteuclid.org/euclid.bjps/1528444869<strong>Zahra Sharifonnasabi</strong>, <strong>Mohammad Hossein Alamatsaz</strong>, <strong>Iraj Kazemi</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 497--524.</p><p><strong>Abstract:</strong><br/>
In this paper, we shall construct a large class of new bivariate copulas. This class happens to contain several known classes of copulas, such as Farlie–Gumbel–Morgenstern, Ali–Mikhail–Haq and Barnett–Gumbel, as its especial members. It is shown that the proposed copulas improve the range of values of correlation coefficient and thus they are more applicable in data modeling. We shall also reveal that the dependent properties of the base copula are preserved by the generated copula under certain conditions. Several members of the new class are introduced as instances and their range of correlation coefficients are computed.
</p>projecteuclid.org/euclid.bjps/1528444869_20180608040117Fri, 08 Jun 2018 04:01 EDTDiagnostics analysis for skew-normal linear regression models: Applications to a quality of life datasethttps://projecteuclid.org/euclid.bjps/1528444870<strong>Clécio da Silva Ferreira</strong>, <strong>Filidor Vilca</strong>, <strong>Heleno Bolfarine</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 525--544.</p><p><strong>Abstract:</strong><br/>
The skew-normal distribution has been used successfully in various statistical applications. The main purpose of this paper is to consider local influence analysis, which is recognized as an important step of data analysis. Motivated to simplify expressions of the conditional expectation of the complete-data log-likelihood function, used in the EM algorithm, diagnostic measures are derived from the case-deletion approach and the local influence approach inspired by Zhu et al. [ Biometrika 88 (2001) 727–737] and Zhu and Lee [ J. R. Stat. Soc. Ser. B. Stat. Methodol. 63 (2001) 111–126]. Finally, the results obtained are applied to a dataset from a study to evaluate quality of life (QOL) and to identify its associated factors in climacteric women with a history of breast cancer.
</p>projecteuclid.org/euclid.bjps/1528444870_20180608040117Fri, 08 Jun 2018 04:01 EDTParameter estimation for discretely observed non-ergodic fractional Ornstein–Uhlenbeck processes of the second kindhttps://projecteuclid.org/euclid.bjps/1528444871<strong>Brahim El Onsy</strong>, <strong>Khalifa Es-Sebaiy</strong>, <strong>Djibril Ndiaye</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 545--558.</p><p><strong>Abstract:</strong><br/>
We use the least squares type estimation to estimate the drift parameter $\theta>0$ of a non-ergodic fractional Ornstein–Uhlenbeck process of the second kind defined as $dX_{t}=\theta X_{t}\,dt+dY_{t}^{(1)},X_{0}=0$, $t\geq0$, where $Y_{t}^{(1)}=\int_{0}^{t}e^{-s}\,dB_{a_{s}}$ with $a_{t}=He^{\frac{t}{H}}$, and $\{B_{t},t\geq0\}$ is a fractional Brownian motion of Hurst parameter $H\in(\frac{1}{2},1)$. We assume that the process $\{X_{t},t\geq0\}$ is observed at discrete time instants $t_{1}=\Delta_{n},\ldots,t_{n}=n\Delta_{n}$. We construct two estimators $\hat{\theta}_{n}$ and $\check{\theta}_{n}$ of $\theta$ which are strongly consistent and we prove that these estimators are $\sqrt{n\Delta_{n}}$-consistent, in the sense that the sequences $\sqrt{n\Delta_{n}}(\hat{\theta}_{n}-\theta)$ and $\sqrt{n\Delta_{n}}(\check{\theta}_{n}-\theta)$ are tight.
</p>projecteuclid.org/euclid.bjps/1528444871_20180608040117Fri, 08 Jun 2018 04:01 EDTA Bayesian approach to errors-in-variables beta regressionhttps://projecteuclid.org/euclid.bjps/1528444872<strong>Jorge Figueroa-Zúñiga</strong>, <strong>Jalmar M. F. Carrasco</strong>, <strong>Reinaldo Arellano-Valle</strong>, <strong>Silvia L. P. Ferrari</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 559--582.</p><p><strong>Abstract:</strong><br/>
Beta regression models have been widely used for the analysis of limited-range continuous variables. Here, we consider an extension of the beta regression models that allows for explanatory variables to be measured with error. Then we propose a Bayesian treatment for errors-in-variables beta regression models. The specification of prior distributions is discussed, computational implementation via Gibbs sampling is provided, and two real data applications are presented. Additionally, Monte Carlo simulations are used to evaluate the performance of the proposed approach.
</p>projecteuclid.org/euclid.bjps/1528444872_20180608040117Fri, 08 Jun 2018 04:01 EDTSums of possibly associated multivariate indicator functions: The Conway–Maxwell-Multinomial distributionhttps://projecteuclid.org/euclid.bjps/1528444873<strong>Joseph B. Kadane</strong>, <strong>Zhi Wang</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 583--596.</p><p><strong>Abstract:</strong><br/>
The Conway–Maxwell-Multinomial distribution is studied in this paper. Its properties are demonstrated, including sufficient statistics and conditions for the propriety of posterior distributions derived from it. An application is given using data from Mendel’s ground-breaking genetic studies.
</p>projecteuclid.org/euclid.bjps/1528444873_20180608040117Fri, 08 Jun 2018 04:01 EDTA note on weak convergence results for infinite causal triangulationshttps://projecteuclid.org/euclid.bjps/1528444874<strong>Valentin Sisko</strong>, <strong>Anatoly Yambartsev</strong>, <strong>Stefan Zohren</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 597--615.</p><p><strong>Abstract:</strong><br/>
We discuss infinite causal triangulations and equivalence to the size biased branching process measure—the critical Galton–Watson branching process distribution conditioned on non-extinction. Using known results from the theory of branching processes, this relation is used to prove a novel weak convergence result of the joint length-area process of a infinite causal triangulations to a limiting diffusion. The diffusion equation enables us to determine the physical Hamiltonian and Green’s function from the Feynman–Kac procedure, providing us with a mathematical rigorous proof of certain scaling limits of causal dynamical triangulations.
</p>projecteuclid.org/euclid.bjps/1528444874_20180608040117Fri, 08 Jun 2018 04:01 EDTSemiparametric quantile estimation for varying coefficient partially linear measurement errors modelshttps://projecteuclid.org/euclid.bjps/1528444875<strong>Jun Zhang</strong>, <strong>Yan Zhou</strong>, <strong>Xia Cui</strong>, <strong>Wangli Xu</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 616--656.</p><p><strong>Abstract:</strong><br/>
We study varying coefficient partially linear models when some linear covariates are error-prone, but their ancillary variables are available. After calibrating the error-prone covariates, we study quantile regression estimates for parametric coefficients and nonparametric varying coefficient functions, and we develop a semiparametric composite quantile estimation procedure. Asymptotic properties of the proposed estimators are established, and the estimators achieve their best convergence rate with proper bandwidth conditions. Simulation studies are conducted to evaluate the performance of the proposed method, and a real data set is analyzed as an illustration.
</p>projecteuclid.org/euclid.bjps/1528444875_20180608040117Fri, 08 Jun 2018 04:01 EDTWeighted sampling without replacementhttps://projecteuclid.org/euclid.bjps/1528444876<strong>Anna Ben-Hamou</strong>, <strong>Yuval Peres</strong>, <strong>Justin Salez</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 657--669.</p><p><strong>Abstract:</strong><br/>
Comparing concentration properties of uniform sampling with and without replacement has a long history which can be traced back to the pioneer work of Hoeffding (1963). The goal of this note is to extend this comparison to the case of non-uniform weights, using a coupling between samples drawn with and without replacement. When the items’ weights are arranged in the same order as their values, we show that the induced coupling for the cumulative values is a submartingale coupling. As a consequence, the powerful Chernoff-type upper-tail estimates known for sampling with replacement automatically transfer to the case of sampling without replacement. For general weights, we use the same coupling to establish a sub-Gaussian concentration inequality. As the sample size approaches the total number of items, the variance factor in this inequality displays the same kind of sharpening as Serfling (1974) identified in the case of uniform weights. We also construct an other martingale coupling which allows us to answer a question raised by Luh and Pippenger (2014) on sampling in Polya urns with different replacement numbers.
</p>projecteuclid.org/euclid.bjps/1528444876_20180608040117Fri, 08 Jun 2018 04:01 EDTOn Hilbert’s 8th problemhttps://projecteuclid.org/euclid.bjps/1528444877<strong>Nicholas G. Polson</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 3, 670--678.</p><p><strong>Abstract:</strong><br/>
A Hadamard factorisation of the Riemann $\xi$-function is constructed to characterize the zeros of the zeta function.
</p>projecteuclid.org/euclid.bjps/1528444877_20180608040117Fri, 08 Jun 2018 04:01 EDTMaxima of branching random walks with piecewise constant variancehttps://projecteuclid.org/euclid.bjps/1534492897<strong>Frédéric Ouimet</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 679--706.</p><p><strong>Abstract:</strong><br/>
This article extends the results of Fang and Zeitouni [ Electron. J. Probab. 17 (2012a) 18] on branching random walks (BRWs) with Gaussian increments in time inhomogeneous environments. We treat the case where the variance of the increments changes a finite number of times at different scales in $[0,1]$ under a slight restriction. We find the asymptotics of the maximum up to an $O_{\mathbb{P}}(1)$ error and show how the profile of the variance influences the leading order and the logarithmic correction term. A more general result was independently obtained by Mallein [ Electron. J. Probab. 20 (2015b) 40] when the law of the increments is not necessarily Gaussian. However, the proof we present here generalizes the approach of Fang and Zeitouni [ Electron. J. Probab. 17 (2012a) 18] instead of using the spinal decomposition of the BRW. As such, the proof is easier to understand and more robust in the presence of an approximate branching structure.
</p>projecteuclid.org/euclid.bjps/1534492897_20180817040200Fri, 17 Aug 2018 04:02 EDTA survival model with Birnbaum–Saunders frailty for uncensored and censored cancer datahttps://projecteuclid.org/euclid.bjps/1534492898<strong>Jeremias Leão</strong>, <strong>Víctor Leiva</strong>, <strong>Helton Saulo</strong>, <strong>Vera Tomazella</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 707--729.</p><p><strong>Abstract:</strong><br/>
Survival models with frailty are used when additional data are non-available to explain the occurrence time of an event of interest. This non-availability may be considered as a random effect related to unobserved explanatory variables, or that cannot be measured, often attributed to environmental or genetic factors. We propose a survival model with frailty based on the Birnbaum–Saunders distribution. This distribution has been widely applied to lifetime data. The random effect is the frailty, which is assumed to follow the Birnbaum–Saunders distribution and introduced on the baseline hazard rate to control the unobservable heterogeneity of the patients. We use the maximum likelihood method to estimate the model parameters and evaluate its performance under different censoring proportions by a Monte Carlo simulation study. Two types of residuals are considered to assess the adequacy of the proposed model. Examples with uncensored and censored real-world data sets illustrate the potential applications of the proposed model.
</p>projecteuclid.org/euclid.bjps/1534492898_20180817040200Fri, 17 Aug 2018 04:02 EDTSearching for the core variables in principal components analysishttps://projecteuclid.org/euclid.bjps/1534492899<strong>Yanina Gimenez</strong>, <strong>Guido Giussani</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 730--754.</p><p><strong>Abstract:</strong><br/>
In this article, we introduce a procedure for selecting variables in principal components analysis. It is developed to identify a small subset of the original variables that best explain the principal components through nonparametric relationships. There are usually some noisy uninformative variables in a dataset, and some variables that are strongly related to one another because of their general dependence. The procedure is designed to be used following the satisfactory initial principal components analysis with all variables, and its aim is to help to interpret the underlying structures. We analyze the asymptotic behavior of the method and provide some examples.
</p>projecteuclid.org/euclid.bjps/1534492899_20180817040200Fri, 17 Aug 2018 04:02 EDTBrazilian network of PhDs working with probability and statisticshttps://projecteuclid.org/euclid.bjps/1534492900<strong>Luciano Digiampietri</strong>, <strong>Leandro Rêgo</strong>, <strong>Filipe Costa de Souza</strong>, <strong>Raydonal Ospina</strong>, <strong>Jesús Mena-Chalco</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 755--782.</p><p><strong>Abstract:</strong><br/>
Statistical and probabilistic reasoning enlightens our judgments about uncertainty and the chance or beliefs on the occurrence of random events in everyday life. Therefore, there are scientists working with Probability and Statistics in various fields of knowledge, what favors the formation of scientific network collaborations of researchers with different backgrounds. Here, we propose to describe the Brazilian PhDs who work with probability and statistics. In particular, we analyze national and states collaboration networks of such researchers by calculating different metrics. We show that there is a greater concentration of nodes in and around the cites which host Probability and Statistics graduate programs. Moreover, the states that host P&S Doctoral programs are the most central. We also observe a disparity in the size of the states networks. The clustering coefficient of the national network suggests that this network and regional differences especially with respect to states from South-east and North is not cohesive and, probably, it is in a maturing stage.
</p>projecteuclid.org/euclid.bjps/1534492900_20180817040200Fri, 17 Aug 2018 04:02 EDTExit time for a reaction diffusion model: Case of a one well potentialhttps://projecteuclid.org/euclid.bjps/1534492901<strong>Adrian Hinojosa</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 783--794.</p><p><strong>Abstract:</strong><br/>
We consider a interacting particle system, the Glauber $+$ Kawasaki model. This model is the result of the combination of a fast stirring, the Kawasaki part, and a spin flip process, the Glauber part. This process has a Reaction–Diffusion equation as hydrodynamic limit, as is proven by De Masi and Presutti ( Mathematical Methods for Hydrodynamic Limits (1991) Springer). The ergodicity of these dynamics (one well potential) was proven in Brasseco et al. ( Amer. Math. Soc. Transl. Ser. 2 198 (2000) 37–49), for any dimension. In this article, we prove the asymptotic exponentiality for certain exit time from a subset of the basin of attraction of the well.
</p>projecteuclid.org/euclid.bjps/1534492901_20180817040200Fri, 17 Aug 2018 04:02 EDTOn the time-dependent Fisher information of a density functionhttps://projecteuclid.org/euclid.bjps/1534492902<strong>Omid Kharazmi</strong>, <strong>Majid Asadi</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 795--814.</p><p><strong>Abstract:</strong><br/>
Fisher information is a very important and fundamental criterion in statistical inference especially in optimal and large sample studies in estimation theory. It also plays a key role in physics, thermodynamic, information theory and other applications. In the literature there have been defined two forms of Fisher information: one for the parameters of a distribution function and one for the density function of a distribution. In this paper, we consider a nonnegative continuous random (lifetime) variable $X$ and define a time-dependent Fisher information for density function of the residual random variable associated to $X$. We also propose a time-dependent version of Fisher information distance (relative Fisher information) between the densities of two nonnegative random variables. Several properties of the proposed measures and their relations to other statistical measures are investigated. To illustrate the results various examples are also provided.
</p>projecteuclid.org/euclid.bjps/1534492902_20180817040200Fri, 17 Aug 2018 04:02 EDTAsymptotic predictive inference with exchangeable datahttps://projecteuclid.org/euclid.bjps/1534492903<strong>Patrizia Berti</strong>, <strong>Luca Pratelli</strong>, <strong>Pietro Rigo</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 815--833.</p><p><strong>Abstract:</strong><br/>
Let $(X_{n})$ be a sequence of random variables, adapted to a filtration $(\mathcal{G}_{n})$, and let $\mu_{n}=(1/n)\sum_{i=1}^{n}\delta_{X_{i}}$ and $a_{n}(\cdot)=P(X_{n+1}\in\cdot|\mathcal{G}_{n})$ be the empirical and the predictive measures. We focus on \begin{equation*}\Vert \mu_{n}-a_{n}\Vert =\mathop{\mathrm{sup}}_{B\in\mathcal{D}}\vert\mu_{n}(B)-a_{n}(B)\vert,\end{equation*} where $\mathcal{D}$ is a class of measurable sets. Conditions for $\Vert \mu_{n}-a_{n}\Vert \rightarrow0$, almost surely or in probability, are given. Also, to determine the rate of convergence, the asymptotic behavior of $r_{n}\Vert \mu_{n}-a_{n}\Vert $ is investigated for suitable constants $r_{n}$. Special attention is paid to $r_{n}=\sqrt{n}$ and $r_{n}=\sqrt{\frac{n}{\log\log n}}$. The sequence $(X_{n})$ is exchangeable or, more generally, conditionally identically distributed.
</p>projecteuclid.org/euclid.bjps/1534492903_20180817040200Fri, 17 Aug 2018 04:02 EDTWavelet estimation for derivative of a density in the presence of additive noisehttps://projecteuclid.org/euclid.bjps/1534492904<strong>B. L. S. Prakasa Rao</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 834--850.</p><p><strong>Abstract:</strong><br/>
We construct a wavelet estimator for the derivative of a probability density function in the presence of an additive noise and study its $L_{p}$-consistency property.
</p>projecteuclid.org/euclid.bjps/1534492904_20180817040200Fri, 17 Aug 2018 04:02 EDTDimension reduction based on conditional multiple index density functionhttps://projecteuclid.org/euclid.bjps/1534492905<strong>Jun Zhang</strong>, <strong>Baohua He</strong>, <strong>Tao Lu</strong>, <strong>Songqiao Wen</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 851--872.</p><p><strong>Abstract:</strong><br/>
In this paper, a dimension reduction method is proposed by using the first derivative of the conditional density function of response given predictors. To estimate the central subspace, we propose a direct methodology by taking expectation of the product of predictor and kernel function about response, which helps to capture the directions in the conditional density function. The consistency and asymptotic normality of the proposed estimation methodology are investigated. Furthermore, we conduct some simulations to evaluate the performance of our proposed method and compare with existing methods, and a real data set is analyzed for illustration.
</p>projecteuclid.org/euclid.bjps/1534492905_20180817040200Fri, 17 Aug 2018 04:02 EDTA weak version of bivariate lack of memory propertyhttps://projecteuclid.org/euclid.bjps/1534492906<strong>Nikolai Kolev</strong>, <strong>Jayme Pinto</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 32, Number 4, 873--906.</p><p><strong>Abstract:</strong><br/>
We suggest a modification of the classical Marshall–Olkin’s bivariate exponential distribution considering a possibility of a singularity contribution along arbitrary line through the origin. It serves as a base of a new weaker version of the bivariate lack of memory property, which might be both “aging” and “non-aging” depending on the additional inclination parameter. The corresponding copula is obtained and we establish its disagreement with Lancaster’s phenomena. Characterizations and properties of the novel bivariate memory-less notion are obtained and its applications are discussed. We characterize associated weak multivariate version. The weak bivariate lack of memory property implies restrictions on the marginal distributions. Starting from pre-specified marginals we propose a procedure to build bivariate distributions possessing a weak bivariate lack of memory property and illustrate it by examples. We complement the methodology with closure properties of the new class. We finish with a discussion and suggest several related problems for future research.
</p>projecteuclid.org/euclid.bjps/1534492906_20180817040200Fri, 17 Aug 2018 04:02 EDTRetraction: On Hilbert’s 8th problemhttps://projecteuclid.org/euclid.bjps/1539361259<strong>Nicholas G. Polson</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 1--1.</p><p><strong>Abstract:</strong><br/>
Two errata in the paper are given.
</p>projecteuclid.org/euclid.bjps/1539361259_20181012122106Fri, 12 Oct 2018 12:21 EDTBimodal extension based on the skew-$t$-normal distributionhttps://projecteuclid.org/euclid.bjps/1547456483<strong>Mehdi Amiri</strong>, <strong>Héctor W. Gómez</strong>, <strong>Ahad Jamalizadeh</strong>, <strong>Mina Towhidi</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 2--23.</p><p><strong>Abstract:</strong><br/>
In this paper, a skew and uni-/bi-modal extension of the Student-$t$ distribution is considered. This model is more flexible and has wider ranges of skewness and kurtosis than the other skew distributions in literature. Fisher information matrix for the proposed model and some submodels are derived. With a simulation study and some real data sets, applicability of the proposed models are illustrated.
</p>projecteuclid.org/euclid.bjps/1547456483_20190114040156Mon, 14 Jan 2019 04:01 ESTExtreme-cum-median ranked set samplinghttps://projecteuclid.org/euclid.bjps/1547456484<strong>Shakeel Ahmed</strong>, <strong>Javid Shabbir</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 24--38.</p><p><strong>Abstract:</strong><br/>
A mixture of Extreme Ranked Set Sampling (ERSS) and Median Ranked Set Sampling (MRSS) is introduced to obtain a more representative sample using three out of five number summary statistics [i.e., Minimum, Median and Maximum]. The proposed sampling scheme provides unbiased estimator of mean for symmetric population and gives moderate efficiency for both symmetric and asymmetric populations under perfect as well as imperfect rankings. Expressions for bias and asymptotic variance are presented. A simulation study is also conducted to observe the performance of the proposed estimator. Application of proposed sampling scheme is illustrated through a real life example.
</p>projecteuclid.org/euclid.bjps/1547456484_20190114040156Mon, 14 Jan 2019 04:01 ESTInventory model of type $(s,S)$ under heavy tailed demand with infinite variancehttps://projecteuclid.org/euclid.bjps/1547456486<strong>Aslı Bektaş Kamışlık</strong>, <strong>Tülay Kesemen</strong>, <strong>Tahir Khaniyev</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 39--56.</p><p><strong>Abstract:</strong><br/>
In this study, a stochastic process $X(t)$, which describes an inventory model of type $(s,S)$ is considered in the presence of heavy tailed demands with infinite variance. The aim of this study is observing the impact of regularly varying demand distributions with infinite variance on the stochastic process $X(t)$. The main motivation of this work is, the publication by Geluk [ Proceedings of the American Mathematical Society 125 (1997) 3407–3413] where he provided a special asymptotic expansion for renewal function generated by regularly varying random variables. Two term asymptotic expansion for the ergodic distribution function of the process $X(t)$ is obtained based on the main results proposed by Geluk [ Proceedings of the American Mathematical Society 125 (1997) 3407–3413]. Finally, weak convergence theorem for the ergodic distribution of this process is proved by using Karamata theory.
</p>projecteuclid.org/euclid.bjps/1547456486_20190114040156Mon, 14 Jan 2019 04:01 ESTExploring the constant coefficient of a single-index variationhttps://projecteuclid.org/euclid.bjps/1547456487<strong>Jun Zhang</strong>, <strong>Cuizhen Niu</strong>, <strong>Gaorong Li</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 57--86.</p><p><strong>Abstract:</strong><br/>
We consider a problem of checking whether the coefficient of the scale and location function is a constant. Both the scale and location functions are modeled as single-index models. Two test statistics based on Kolmogorov–Smirnov and Cramér–von Mises type functionals of the difference of the empirical residual processes are proposed. The asymptotic distribution of the estimator for single-index parameter is derived, and the empirical distribution function of residuals is shown to converge to a Gaussian process. Moreover, the proposed test statistics can be able to detect local alternatives that converge to zero at a parametric convergence rate. A bootstrap procedure is further proposed to calculate critical values. Simulation studies and a real data analysis are conducted to demonstrate the performance of the proposed methods.
</p>projecteuclid.org/euclid.bjps/1547456487_20190114040156Mon, 14 Jan 2019 04:01 ESTTransdimensional transformation based Markov chain Monte Carlohttps://projecteuclid.org/euclid.bjps/1547456488<strong>Moumita Das</strong>, <strong>Sourabh Bhattacharya</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 87--138.</p><p><strong>Abstract:</strong><br/>
Variable dimensional problems, where not only the parameters, but also the number of parameters are random variables, pose serious challenge to Bayesians. Although in principle the Reversible Jump Markov Chain Monte Carlo (RJMCMC) methodology is a response to such challenges, the dimension-hopping strategies need not be always convenient for practical implementation, particularly because efficient “move-types” having reasonable acceptance rates are often difficult to devise.
In this article, we propose and develop a novel and general dimension-hopping MCMC methodology that can update all the parameters as well as the number of parameters simultaneously using simple deterministic transformations of some low-dimensional (often one-dimensional) random variable. This methodology, which has been inspired by Transformation based MCMC (TMCMC) of ( Stat. Mehodol. (2014) 16 100–116), facilitates great speed in terms of computation time and provides reasonable acceptance rates and mixing properties. Quite importantly, our approach provides a natural way to automate the move-types in variable dimensional problems. We refer to this methodology as Transdimensional Transformation based Markov Chain Monte Carlo (TTMCMC). Comparisons with RJMCMC in gamma and normal mixture examples demonstrate far superior performance of TTMCMC in terms of mixing, acceptance rate, computational speed and automation. Furthermore, we demonstrate good performance of TTMCMC in multivariate normal mixtures, even for dimension as large as $20$. To our knowledge, there exists no application of RJMCMC for such high-dimensional mixtures.
As by-products of our effort on the development of TTMCMC, we propose a novel methodology to summarize the posterior distributions of the mixture densities, providing a way to obtain the mode of the posterior distribution of the densities and the associated highest posterior density credible regions. Based on our method, we also propose a criterion to assess convergence of variable-dimensional algorithms. These methods of summarization and convergence assessment are applicable to general problems, not just to mixtures.
</p>projecteuclid.org/euclid.bjps/1547456488_20190114040156Mon, 14 Jan 2019 04:01 ESTBootstrap for correcting the mean square error of prediction and smoothed estimates in structural modelshttps://projecteuclid.org/euclid.bjps/1547456490<strong>Thiago R. dos Santos</strong>, <strong>Glaura C. Franco</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 139--160.</p><p><strong>Abstract:</strong><br/>
It is well known that the uncertainty in the estimation of parameters produces the underestimation of the mean square error (MSE) both for in-sample and out-of-sample estimation. In the state space framework, this problem can affect confidence intervals for smoothed estimates and forecasts, which are generally built by state vector predictors that use estimated model parameters. In order to correct this problem, this paper proposes and compares parametric and nonparametric bootstrap methods based on procedures usually employed to calculate the MSE in the context of forecasting and smoothing in state space models. The comparisons are performed through an extensive Monte Carlo study which illustrates, empirically, the bias reduction in the estimation of MSE for prediction and smoothed estimates using the bootstrap approaches. The finite sample properties of the bootstrap procedures are analyzed for Gaussian and non-Gaussian assumptions of the error term. The procedures are also applied to real time series, leading to satisfactory results.
</p>projecteuclid.org/euclid.bjps/1547456490_20190114040156Mon, 14 Jan 2019 04:01 ESTFitting mixed models to messy longitudinal data: A case study involving estimation of post mortem intervalshttps://projecteuclid.org/euclid.bjps/1547456491<strong>Julio M. Singer</strong>, <strong>Francisco M. M. Rocha</strong>, <strong>Carmen D. S. André</strong>, <strong>Talita Zerbini</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 161--183.</p><p><strong>Abstract:</strong><br/>
Non-linear mixed models are useful in many practical longitudinal data problems, especially when they are derived as solutions to differential equations generated by subject matter theoretical considerations. When this underlying rationale is not available, practitioners are faced with the dilemma of choosing a model from the numerous ones available in the literature. The situation is even worse for messy data where interpretation and computational problems are frequent. This is the case with a pilot observational study conducted at the School of Medicine of the University of São Paulo in which a new method to estimate the time since death (post-mortem interval—PMI) is proposed. In particular, the attenuation of the density of intra-cardiac hypostasis (concentration of red cells in the vascular system by gravity) obtained from a series of tomographic images was observed in the thoraces of 21 bodies of hospitalized patients with known time of death. The images were obtained at different instants and not always at the same conditions for each body, generating a set of messy data. In this context, we consider three ad hoc models to analyse the data, commenting on the advantages and caveats of each approach.
</p>projecteuclid.org/euclid.bjps/1547456491_20190114040156Mon, 14 Jan 2019 04:01 ESTThe equivalence of dynamic and static asset allocations under the uncertainty caused by Poisson processeshttps://projecteuclid.org/euclid.bjps/1547456492<strong>Yong-Chao Zhang</strong>, <strong>Na Zhang</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 184--191.</p><p><strong>Abstract:</strong><br/>
We investigate the equivalence of dynamic and static asset allocations in the case where the price process of a risky asset is driven by a Poisson process. Under some mild conditions, we obtain a necessary and sufficient condition for the equivalence of dynamic and static asset allocations. In addition, we provide a simple sufficient condition for the equivalence.
</p>projecteuclid.org/euclid.bjps/1547456492_20190114040156Mon, 14 Jan 2019 04:01 ESTSimple tail index estimation for dependent and heterogeneous data with missing valueshttps://projecteuclid.org/euclid.bjps/1547456493<strong>Ivana Ilić</strong>, <strong>Vladica M. Veličković</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 1, 192--203.</p><p><strong>Abstract:</strong><br/>
Financial returns are known to be nonnormal and tend to have fat-tailed distribution. Also, the dependence of large values in a stochastic process is an important topic in risk, insurance and finance. In the presence of missing values, we deal with the asymptotic properties of a simple “median” estimator of the tail index based on random variables with the heavy-tailed distribution function and certain dependence among the extremes. Weak consistency and asymptotic normality of the proposed estimator are established. The estimator is a special case of a well-known estimator defined in Bacro and Brito [ Statistics & Decisions 3 (1993) 133–143]. The advantage of the estimator is its robustness against deviations and compared to Hill’s, it is less affected by the fluctuations related to the maximum of the sample or by the presence of outliers. Several examples are analyzed in order to support the proofs.
</p>projecteuclid.org/euclid.bjps/1547456493_20190114040156Mon, 14 Jan 2019 04:01 ESTBayesian robustness to outliers in linear regression and ratio estimationhttps://projecteuclid.org/euclid.bjps/1551690032<strong>Alain Desgagné</strong>, <strong>Philippe Gagnon</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 205--221.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690032_20190304040045Mon, 04 Mar 2019 04:00 ESTA brief review of optimal scaling of the main MCMC approaches and optimal scaling of additive TMCMC under non-regular caseshttps://projecteuclid.org/euclid.bjps/1551690033<strong>Kushal K. Dey</strong>, <strong>Sourabh Bhattacharya</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 222--266.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690033_20190304040045Mon, 04 Mar 2019 04:00 ESTThe coreset variational Bayes (CVB) algorithm for mixture analysishttps://projecteuclid.org/euclid.bjps/1551690034<strong>Qianying Liu</strong>, <strong>Clare A. McGrory</strong>, <strong>Peter W. J. Baxter</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 267--279.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690034_20190304040045Mon, 04 Mar 2019 04:00 ESTModified information criterion for testing changes in skew normal modelhttps://projecteuclid.org/euclid.bjps/1551690035<strong>Khamis K. Said</strong>, <strong>Wei Ning</strong>, <strong>Yubin Tian</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 280--300.</p><p><strong>Abstract:</strong><br/>
In this paper, we study the change point problem for the skew normal distribution model from the view of model selection problem. The detection procedure based on the modified information criterion (MIC) for change problem is proposed. Such a procedure has advantage in detecting the changes in early and late stage of a data comparing to the one based on the traditional Schwarz information criterion which is well known as Bayesian information criterion (BIC) by considering the complexity of the models. Due to the difficulty in deriving the analytic asymptotic distribution of the test statistic based on the MIC procedure, the bootstrap simulation is provided to obtain the critical values at the different significance levels. Simulations are conducted to illustrate the comparisons of performance between MIC, BIC and likelihood ratio test (LRT). Such an approach is applied on two stock market data sets to indicate the detection procedure.
</p>projecteuclid.org/euclid.bjps/1551690035_20190304040045Mon, 04 Mar 2019 04:00 ESTFailure rate of Birnbaum–Saunders distributions: Shape, change-point, estimation and robustnesshttps://projecteuclid.org/euclid.bjps/1551690036<strong>Emilia Athayde</strong>, <strong>Assis Azevedo</strong>, <strong>Michelli Barros</strong>, <strong>Víctor Leiva</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 301--328.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690036_20190304040045Mon, 04 Mar 2019 04:00 ESTA new log-linear bimodal Birnbaum–Saunders regression model with application to survival datahttps://projecteuclid.org/euclid.bjps/1551690037<strong>Francisco Cribari-Neto</strong>, <strong>Rodney V. Fonseca</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 329--355.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690037_20190304040045Mon, 04 Mar 2019 04:00 ESTNecessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walkhttps://projecteuclid.org/euclid.bjps/1551690038<strong>Bastien Mallein</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 356--373.</p><p><strong>Abstract:</strong><br/>
Consider a supercritical branching random walk on the real line. The consistent maximal displacement is the smallest of the distances between the trajectories followed by individuals at the $n$th generation and the boundary of the process. Fang and Zeitouni, and Faraud, Hu and Shi proved that under some integrability conditions, the consistent maximal displacement grows almost surely at rate $\lambda^{*}n^{1/3}$ for some explicit constant $\lambda^{*}$. We obtain here a necessary and sufficient condition for this asymptotic behaviour to hold.
</p>projecteuclid.org/euclid.bjps/1551690038_20190304040045Mon, 04 Mar 2019 04:00 ESTHierarchical modelling of power law processes for the analysis of repairable systems with different truncation times: An empirical Bayes approachhttps://projecteuclid.org/euclid.bjps/1551690039<strong>Rodrigo Citton P. dos Reis</strong>, <strong>Enrico A. Colosimo</strong>, <strong>Gustavo L. Gilardoni</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 374--396.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690039_20190304040045Mon, 04 Mar 2019 04:00 ESTA temporal perspective on the rate of convergence in first-passage percolation under a moment conditionhttps://projecteuclid.org/euclid.bjps/1551690040<strong>Daniel Ahlberg</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 397--401.</p><p><strong>Abstract:</strong><br/>
We study the rate of convergence in the celebrated Shape Theorem in first-passage percolation, obtaining the precise asymptotic rate of decay for the probability of linear order deviations under a moment condition. Our results are presented from a temporal perspective and complement previous work by the same author, in which the rate of convergence was studied from the standard spatial perspective.
</p>projecteuclid.org/euclid.bjps/1551690040_20190304040045Mon, 04 Mar 2019 04:00 ESTInfluence measures for the Waring regression modelhttps://projecteuclid.org/euclid.bjps/1551690041<strong>Luisa Rivas</strong>, <strong>Manuel Galea</strong>. <p><strong>Source: </strong>Brazilian Journal of Probability and Statistics, Volume 33, Number 2, 402--424.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bjps/1551690041_20190304040045Mon, 04 Mar 2019 04:00 EST