A Note on Priors for the Multinomial Model

An “overall objective” prior proposed for the multinomial model is shown to be inadequate in the presence of zero counts. An earlier proposed reference prior for when interest is in a particular category suffers from similar problems. It is argued that there is no need to deviate from the uniform prior proposed by Jeffreys, for which links with a non-Bayesian approach, when prediction is of interest, are shown.     

Bayesian inference for Birnbaum–Saunders distribution and its generalization

We present a Bayesian approach for parameter inference of the Birnbaum–Saunders distribution [Birnbaum ZW, Saunders SC. A new family of life distributions. J Appl Probab. 1969;6:319–327], as well as the generalized Birnbaum–Saunders distribution developed by Owen [A new three-parameter extension to the Birnbaum–Saunders distribution. IEEE Trans Reliab. 2006;55:475–479], in the presence of random right-censored data. To handle the instance of commonly occurred censored observations, we utilize the data augmentation technique [Tanner MA, Wong WH. The calculation of posterior distributions by data augmentation. J Amer Statist Assoc. 1987;82(398):528–540] to circumvent the arduous expressions involving the censored data in posterior inferences. Simulation studies are carried out to assess performance of these methods under different parameter values, with small and large sample sizes, as well as various degrees of censoring. Two real data are analysed for illustrative purpose.     

Optimal trajectory to the general equilibrium state of the unstable stochastic lattice gas of prey–predator model with pair-approximation

This paper focuses on the study of the optimal control problem of the unstable stochastic lattice gas of prey–predator model with pair-approximation. The Pontryagin minimum procedure is used to derive the optimal behaviour of the proposed system that makes it trajectory closer to its general equilibrium states with respect to a selected performance measure during a specified period of time T. Wide numerical simulations and illustrations are presented in graphs and tables.     

The shapes of things to come: probability density quantiles

For every discrete or continuous location-scale family having a square-integrable density, there is a unique continuous probability distribution on the unit interval that is determined by the density-quantile composition introduced by Parzen in 1979. These probability density quantiles (pdQs) only differ in shape, and can be usefully compared with the Hellinger distance or Kullback–Leibler divergences. Convergent empirical estimates of these pdQs are provided, which leads to a robust global fitting procedure of shape families to data. Asymmetry can be measured in terms of distance or divergence of pdQs from the symmetric class. Further, a precise classification of shapes by tail behaviour can be defined simply in terms of pdQ boundary derivatives.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.     

Estimation of the bivariate distribution function for censored gap times

In many medical studies, patients may experience several events during follow-up. The times between consecutive events (gap times) are often of interest and lead to problems that have received much attention recently. In this work, we consider the estimation of the bivariate distribution function for censored gap times. Some related problems such as the estimation of the marginal distribution of the second gap time and the conditional distribution are also discussed. In this article, we introduce a nonparametric estimator of the bivariate distribution function based on Bayes’ theorem and Kaplan–Meier survival function and explore the behavior of the four estimators through simulations. Real data illustration is included.     

A Monte Carlo-based pseudo-coefficient of determination for generalized linear models with binary outcome

In this article, we focus on a pseudo-coefficient of determination for generalized linear models with binary outcome. Although there are numerous coefficients of determination proposed in the literature, none of them is identified as the best in terms of estimation accuracy, or incorporates all desired characteristics of a precise coefficient of determination. Considering this, we propose a new coefficient of determination by using a computational Monte Carlo approach, and exhibit main characteristics of the proposed coefficient of determination both analytically and numerically. We evaluate and compare performances of the proposed and nine existing coefficients of determination by a comprehensive Monte Carlo simulation study. The proposed measure is found superior to the existent measures when dependent variable is balanced or moderately unbalanced for probit, logit, and complementary log–log link functions and a wide range of sample sizes. Due to the extensive design space of our simulation study, we identify new conditions in which previously recommended coefficients of determination should be used carefully.