ISSN (0103-0752), Vol. 30, No. 2 -- May 2016

# Brazilian Journal of Probability and Statistics

AN OFFICIAL JOURNAL OF THE
INSTITUTE OF MATHEMATICAL STATISTICS

## Articles

 A semiparametric Bayesian model for multiple monotonically increasing count sequences Valeria Leiva-Yamaguchi, Fernando A. Quintana 155-170 Parametric Stein operators and variance bounds Christophe Ley, Yvik Swan 171-195 Log-symmetric distributions: Statistical properties and parameter estimation Luis Hernando Vanegas, Gilberto A. Paula 196-220 Local limit theorems for shock models Edward Omey, Rein Vesilo 221-247 A new skew logistic distribution: Properties and applications D. V. S. Sastry, Deepesh Bhati 248-271 Fractional absolute moments of heavy tailed distributions Muneya Matsui, Zbyněk Pawlas 272-298 Limiting behavior of the Jeffreys power-expected-posterior Bayes factor in Gaussian linear models D. Fouskakis, I. Ntzoufras 299-320 On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$ Jiaojiao Ma, Shengjun Fan, Rui Fang 321-344

## A semiparametric Bayesian model for multiple monotonically increasing count sequences

Valeria Leiva-Yamaguchi, Fernando A. Quintana, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 155-170

### Abstract

In longitudinal clinical trials, subjects may be evaluated many times over the course of the study. This article is motivated by a medical study conducted in the U.S. Veterans Administration Cooperative Urological Research Group to assess the effectiveness of a treatment in preventing recurrence on subjects affected by bladder cancer. The data consist of the accumulated tumor counts over a sequence of regular checkups, with many missing observations. We propose a hierarchical nonparametric Bayesian model for sequences of monotonically increasing counts. Unlike some of the previous analyses for these data, we avoid interpolation by explicitly incorporating the missing observations under the assumption of these being missing completely at random. Our formulation involves a generalized linear mixed effects model, using a dependent Dirichlet process prior for the random effects, with an autoregressive component to include serial correlation along patients. This provides great flexibility in the desired inference, that is, assessing the treatment effect. We discuss posterior computations and the corresponding results obtained for the motivating dataset, including a comparison with parametric alternatives.

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## Parametric Stein operators and variance bounds

Christophe Ley, Yvik Swan, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 171-195

### Abstract

Stein operators are (differential/difference) operators which arise within the so-called Stein’s method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric distributions with continuous dependence on the parameter. We provide explicit general expressions for location, scale and skewness families. We also provide a general expression for discrete distributions. We use properties of our operators to provide upper and lower variance bounds (only lower bounds in the discrete case) on functionals $h(X)$ of random variables $X$ following parametric distributions. These bounds are expressed in terms of the first two moments of the derivatives (or differences) of $h$. We provide general variance bounds for location, scale and skewness families and apply our bounds to specific examples (namely the Gaussian, exponential, gamma and Poisson distributions). The results obtained via our techniques are systematically competitive with, and sometimes improve on, the best bounds available in the literature.

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## Log-symmetric distributions: Statistical properties and parameter estimation

Luis Hernando Vanegas, Gilberto A. Paula, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 196-220

### Abstract

In this paper, we study the main statistical properties of the class of log-symmetric distributions, which includes as special cases bimodal distributions as well as distributions that have heavier/lighter tails than those of the log-normal distribution. This family includes distributions such as the log-normal, log-Student-$t$, harmonic law, Birnbaum–Saunders, Birnbaum–Saunders-$t$ and generalized Birnbaum–Saunders. We derive quantile-based measures of location, dispersion, skewness, relative dispersion and kurtosis for the log-symmetric class that are appropriate in the context of asymmetric and heavy-tailed distributions. Additionally, we discuss parameter estimation based on both classical and Bayesian approaches. The usefulness of the log-symmetric class is illustrated through a statistical analysis of a real dataset, in which the performance of the log-symmetric class is compared with that of some competitive and very flexible distributions.

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## Local limit theorems for shock models

Edward Omey, Rein Vesilo, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 221-247

### Abstract

In many physical systems, failure occurs when the stress after shock $n$ first exceed a critical level $x$. We consider the number of shocks $\tau(x)$ to failure and obtain more detailed information that is usually obtained about asymptotic distribution by using local limit theorems. We consider extreme and cumulative shock models with both univariate and multivariate shock types. We derive the limiting distribution of $\tau(x)$ and the rate of convergence to that. For the extreme shock model, rate of convergence for regularly varying shock distributions is found using the weighted Kolmorogov probability metric. For the cumulative shock model, we examine the rate of convergence to Gaussian densities.

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## A new skew logistic distribution: Properties and applications

D. V. S. Sastry, Deepesh Bhati, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 248-271

### Abstract

Following the methodology of Azzalini, researchers have developed skew logistic distribution and studied its properties. The cumulative distribution function in their case is not explicit and therefore numerical methods are employed for estimation of parameters. In this paper, we develop a new skew logistic distribution based on the methodology of Fernández and Steel and derive its cumulative distribution function and also the characteristic function. For estimating the parameters, Method of Moments, Modified Method of Moment and Maximum likelihood estimation are used. With the help of simulation study, for different sample sizes, the parameters are estimated and their consistency was verified through Box Plot. We also proposed a regression model in which probability of occurrence of an event is derived from our proposed new skew logistic distribution. Further, proposed model fitted to a well studied lean body mass of Australian athlete data and compared with other available competing distributions.

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## Fractional absolute moments of heavy tailed distributions

Muneya Matsui, Zbyněk Pawlas, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 272-298

### Abstract

Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. Our main focus is on an infinite variance case with finite mean, that is, we are interested in formulae for $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$ with $1<\gamma<2$ and $\mu\in\mathbb{R}$. We review techniques of fractional differentiation of Laplace transforms and characteristic functions. Several examples are given with analytical expressions of $\mathbb{E} [\vert X-\mu\vert^{\gamma}]$. We also evaluate the fractional moment errors for both prediction and parameter estimation problems.

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## Limiting behavior of the Jeffreys power-expected-posterior Bayes factor in Gaussian linear models

D. Fouskakis, I. Ntzoufras, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 299-320

### Abstract

Expected-posterior priors (EPPs) have been proved to be extremely useful for testing hypotheses on the regression coefficients of normal linear models. One of the advantages of using EPPs is that impropriety of baseline priors causes no indeterminacy in the computation of Bayes factors. However, in regression problems, they are based on one or more training samples, that could influence the resulting posterior distribution. On the other hand, the power-expected-posterior priors are minimally-informative priors that reduce the effect of training samples on the EPP approach, by combining ideas from the power-prior and unit-information-prior methodologies. In this paper, we prove the consistency of the Bayes factors when using the power-expected-posterior priors, with the independence Jeffreys as a baseline prior, for normal linear models, under very mild conditions on the design matrix.

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## On the stability theorem of $L^{p}$ solutions for multidimensional BSDEs with uniform continuity generators in $z$

Jiaojiao Ma, Shengjun Fan, Rui Fang, Braz. J. Probab. Stat., vol. 30, iss. 2 (2016), 321-344

### Abstract

In this paper, we first establish an existence and uniqueness result of $L^{p}$ ($p>1$) solutions for multidimensional backward stochastic differential equations (BSDEs) whose generator $g$ satisfies a certain one-sided Osgood condition with a general growth in $y$ as well as a uniform continuity condition in $z$, and the $i$th component ${}^{i}g$ of $g$ depends only on the $i$th row ${}^{i}z$ of matrix $z$ besides $(\omega,t,y)$. Then we put forward and prove a stability theorem for $L^{p}$ solutions of this kind of multidimensional BSDEs. This generalizes some known results.

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