We consider the problem of simultaneous variable selection and estimation in partially linear additive models with a large number of grouped variables in the linear part and a large number of nonparametric components. In our problem, the number of grouped variables may be larger than the sample size, but the number of important groups is “small” relative to the sample size. We apply the adaptive group Lasso to select the important groups, using spline bases to approximate the nonparametric components and the group Lasso to obtain an initial consistent estimator. Under appropriate conditions, it is shown that, the group Lasso selects the number of groups which is comparable with the underlying important groups and is estimation consistent, the adaptive group Lasso selects the correct important groups with probability converging to one as the sample size increases and is selection consistent. The results of simulation studies show that the adaptive group Lasso procedure works well with samples of moderate size. A real example is used to illustrate the application of the proposed penalized method.

A full Bayesian method utilizing data augmentation and Gibbs sampling algorithms is presented for analyzing nonignorable missing data. The discussion focuses on a simplified selection model for regression analysis. Regardless of missing mechanisms, it is assumed that missingness only depends on the missing variable itself. Simulation results demonstrate that the simplified selection model can recover regression model parameters under both correctly specified situations and many misspecified situations. The method is also applied to analyzing a training intervention data set with missing data.

We mainly investigate certain mixtures of Poisson distributions with a scale parameter in the mixing distribution. They help us to derive the bivariate Poisson mixtures arising from the prior and posterior predictive distributions in the semi-conjugate family defined by Laurent and Legrand (ESAIM Probab. Stat. (2011) DOI:10.1051/ps/2010018) for the “two Poisson samples” model, which contains in particular the reference prior when the parameter of interest is the ratio of the two Poisson rates.

The usual Bayesian approach for count data is Gamma/Poisson conjugate analysis. However, in this conjugate analysis the influence of the prior distribution could be dominant even when prior and likelihood are in conflict. Our proposal is an analysis based on the Cauchy prior for natural parameter in exponential families. In this work, we show that the Cauchy/Poisson posterior model is a robust model for count data in contrast with the usual conjugate Bayesian approach Gamma/Poisson model. We use the polynomial tails comparison theorem given in (Bayesian Anal. 4 (2009) 817–843) that gives easy-to-check conditions to ensure prior robustness. In short, this means that when the location of the prior and the bulk of the mass of the likelihood get further apart (a situation of conflict between prior and likelihood information), Bayes theorem will cause the posterior distribution to discount the prior information. Finally, we analyze artificial data sets to investigate the robustness of the Cauchy/Poisson model.

We show that all multivariate extreme value distributions, which are the possible weak limits of the K largest order statistics of i.i.d. samples, have the same copula, the so called K-extremal copula. This copula and its density are described through exact expressions. We also study measures of dependence, we obtain a weak convergence result and we propose a simulation algorithm for the K-extremal copula.

Zero-inflated Poisson (ZIP) models, which are mixture models, have been popularly used for count data that often contain large numbers of zeros, but their identifiability has not yet been thoroughly explored. In this work, we systematically investigate the identifiability of the ZIP models under a number of different assumptions. More specifically, we show the identifiability of a parametric ZIP model in which the incidence probability p(x) and Poisson mean λ(x) are modeled parametrically as p(x) = exp(β₀ + β₁x)/[1 + exp(β₀ + β₁x)] and λ(x) = exp(α₀+α₁x) for x being a continuous covariate in a closed interval. A semiparametric ZIP regression model is shown to be identifiable in which (i) p(x) = exp(β₀ + β₁x)/[1 + exp(β₀ + β₁x)] and λ(x) = exp[s(x)], (ii) p(x) = exp[r(x)]/{1 + exp[r(x)]} and λ(x) = exp(α₀ + α₁x), or (iii) p(x) = exp[r(x)]/{1 + exp[r(x)]} and λ(x) = exp[s(x)] for r(x) and s(x) being unspecified smooth functions.

Long-term survival models have historically been considered for analyzing time-to-event data with long-term survivors fraction. However, situations in which a fraction (1 − p) of systems is subject to failure from independent competing causes of failure, while the remaining proportion p is cured or has not presented the event of interest during the time period of the study, have not been fully considered in the literature. In order to accommodate such situations, we present in this paper a new long-term survival model. Maximum likelihood estimation procedure is discussed as well as interval estimation and hypothesis tests. A real dataset illustrates the methodology.