In this paper we propose a criterion to estimate the density function by means of a nonparametric and fuzzy set estimator, based on $n$ i.i.d random variable, obtaining a reduction of the integrated mean square error of the fuzzy set estimator regarding the integrated mean squared error of the classic kernel estimators. This reduction shows that the fuzzy set estimator has better performance than the kernel estimations. Also, the convergence rate of the optimal scaling factor is computed, which coincides with the convergence rate in classic kernel estimation. Finally, these theoretical findings are illustrated using a numerical example.

A new continuous distribution, so-called the beta log-logistic distribution, that extends the log-logistic distribution and some other distributions is proposed and studied. The new model is quite flexible to analyze positive data. Various structural properties of the new distribution are derived, including explicit expressions for the moments, mean deviations and Rényi and Shannon entropies. The score function is derived and the estimation of the model parameters is performed by maximum likelihood. We also determine the expected information matrix. The usefulness of the new model is illustrated by means of two real data sets. We hope that the new distribution proposed here will serve as an alternative model to other models available in the literature for modeling positive real data in many areas.

We propose a model for lifetime data in which all units start out susceptible to the event of interest but may move into a non-susceptible group if another event intervenes. Practical examples include subjects dropping out of a study by leaving the study area. It is supposed that the investigator is unaware of each subject’s status. The model results in the appearance of a cured fraction (long-term survivors). It is shown that certain parametric models for the lifetime and the intervening event are related to a cured fraction mixture model even though the non-susceptible group is fixed from the outset in the latter but not in the new model. A likelihood ratio test and a diagnostic plot are proposed and examples of applications are provided.

For the analysis of square contingency tables, we propose the Kullback–Leibler information type measure to represent the degree of departure from the quasi-symmetry (QS) model. We introduce the global quasi-symmetry (GQS) model, and show that the QS model holds if and only if both the GQS and extended quasi-symmetry (EQS) models hold. Furthermore, we propose a measure of departure from each of the GQS and the EQS models, and show that the value of measure of QS is equal to the sum of the value of measure of GQS and that of EQS.

We study a rumor model from a percolation theory and a branching process point of view. The existence of a giant component is related to the event where a rumor spreads out trough an infinite number of individuals. We present sharp lower and upper bounds for the probability of that event, according to the distribution of the random variables defining the radius of influence of each individual.

We give the spectral decomposition and inverse of multinomial and negative multinomial covariances and related matrices.

To estimate the proportion of a sensitive attribute of the population that is composed of the number of different sized clusters, we suggest a two-stage randomized response model with unequal probability sampling by using Abdelfatah et al.’s procedure [Braz. J. Probab. Stat. 27 (2013) 608–617]. We compute the estimate of the sensitive parameter, its variance, and the variance estimator for both pps sampling and two-stage equal probability sampling. We extend our model to the case of stratified unequal probability sampling and compute them. Finally, we compare the efficiency of the two estimators, one obtained by unequal probability sampling and the other by stratified unequal probability sampling.

Bivariate Schur-constant distributions have an important role in Bayesian analysis of lifetime data, as models possessing no-ageing property. In the present work, we obtain characterizations of bivariate Schur-constant distributions by properties of functions of random variables and reliability concepts. Various time-dependent measures are analysed and shown to be characterized by the ageing property of the marginal distribution.

The main result is a bivariate central limit theorem for “rectangular” sums of dependent complex-valued random variables, indexed by $\mathbb{Z} ^{d}$, which are $\rho^{\prime}$-mixing. An interesting corollary concerning the limiting behavior of the moments of the sums is then proved.

Species richness is related to various factors, such as mutation, interaction, competition and the amount of available resources for survival, among others. Additionally, species richness is related to the size of the habitat area (species-area relationship). A major problem in ecology is to identify the best function that models this relationship. The main aim of the paper is to propose a generalized fitting formula for the species-area relationship. The advantage of our model is that it leads to a unique formulation for species-area relationships, which takes into account both the effects of the minimal area and asymptotic behavior of the growth curve for large areas. This approach provides a unique algorithm for fitting different datasets, and choosing the best model in light of the data. The applicability of our approach was tested via a simulation study conducted to determine if two usual selection criteria are suitable to choose the best formulation to be considered to describe species-area relationships. Real datasets involving a species richness of fish in $70$ lakes was analysed using the proposed methodology.