We introduce a Bayesian analysis for beta generalized distributions and related exponentiated models. We review the exponentiated exponential, exponentiated Weibull and beta generalized exponential distributions. These distributions have been proposed as alternative extensions of the gamma and Weibull distributions in the analysis of lifetime data. Some posterior summaries of interest are obtained using Monte Carlo Markov chain (MCMC) methods. An application to a real data set is given to illustrate the potentiality of the Bayesian analysis.

Let $\{X,X_{i},i\geq 1\}$ be a sequence of independent and identically distributed positive random variables with $E(X)=\mu >0$, $\operatorname{Var}(X)<\infty$. Put $S_{n}=\sum_{i=1}^{n}X_{i}$ and let $g(x)$ be a positive and differentiable function defined on $(0,+\infty)$ satisfying some mild conditions. We prove that, for any $s>1$, \[\lim_{\varepsilon\rightarrow0}\varepsilon^{1/s}\sum_{n=1}^{\infty}g'(n)P\Biggl\{\Biggl|\log\Biggl(\prod_{j=1}^{n}\frac{S_{j}}{j\mu}\Biggr)\Biggr|\geq\varepsilon\sqrt{n}g^{s}(n)\Biggr\}=E|N|^{1/s},\] where $N$ is a standard normal random variable. This result was also extended to product of U-statistics.

The paper by Kumaraswamy (Journal of Hydrology 46 (1980) 79–88) introduced a probability distribution for double bounded random processes which has considerable attention in hydrology and related areas. Based on this distribution, we propose a generalization of the Kumaraswamy distribution refereed to as the exponentiated Kumaraswamy distribution. We derive the moments, moment generating function, mean deviations, Bonferroni and Lorentz curves, density of the order statistics and their moments. We also present a related distribution, so-called the log-exponentiated Kumaraswamy distribution, which extends the generalized exponential (Aust. N. Z. J. Stat. 41 (1999) 173–188) and double generalized exponential (J. Stat. Comput. Simul. 80 (2010) 159–172) distributions. We discuss maximum likelihood estimation of the model parameters. In applications to real data sets, we show that the log-exponentiated Kumaraswamy model can be used quite effectively in analyzing lifetime data.

Consider a Voronoi tiling of $\mathbb{R} ^{d}$ based on a realization of an inhomogeneous Poisson random set. A Voronoi polyomino is a finite and connected union of Voronoi tiles. In this paper we provide tail bounds for the number of boxes that are intersected by a Voronoi polyomino, and vice-versa. These results will be crucial to analyze self-avoiding paths, greedy polyominoes and first-passage percolation models on Voronoi tilings and on the dual graph, named the Delaunay triangulation [Asymptotics for first-passage times on Delaunay triangulations (2011) Preprint, Greedy Polyominoes and first-passage times on random Voronoi tilings (2012) Preprint].

This paper describes a new statistical analysis strategy to problems of cytogenetic dosimetry involving ordinal polythomous responses. Models relating the multivariate response to dose take the data ordinality into account and are analysed in a fully Bayesian fashion in the application here considered. In particular, these models are compared in order to select the best one for purposes of drawing inferences of interest and dose prediction is naturally addressed by its practical importance. This work was motivated by an in vitro experimental study on radiation exposure of human blood cell cultures, previously analysed in the literature by other methods, but its interest holds in many other applications of the biological and environmental field involving data sets yielded from the same type of assays for genetic damage.

In this paper we introduce a new method to add a parameter to a family of distributions. The additional parameter is completely studied and a full description of its behaviour in the distribution is given. We obtain several mathematical properties of the new class of distributions such as Kullback–Leibler divergence, Shannon entropy, moments, order statistics, estimation of the parameters and inference for large sample. Further, we show that the new distribution has the reference distribution as special case, and that the usual inference procedures also hold in this case. We present a comprehensive study of two special cases of the exp-$G$ class: exp-Weibull and exp-beta distributions. Further, an application to the real data set is presented. This family also opens a wide variety of research, as the authors may develop its special cases in full detail.

Azzalini’s skew-normal distribution is obtained through a conditional reduction of a multivariate normal distribution parameterized with a correlation matrix. It seems natural that when the parameterization of that multivariate normal distribution is complexified, more flexible skew-normal distributions could be obtained. In this note this specification strategy, previously explored by Azzalini [Scand. J. Stat. 33 (2006) 561–574] among many other authors, is formally analyzed through an identification analysis.