An approximation of logarithmic functions in the regression setting

We consider a method of moments approach for dealing with censoring at zero for data expressed in levels when researchers would like to take logarithms. A Box–Cox transformation is employed. We explore this approach in the context of linear regression where both dependent and independent variables are censored. We contrast this method to two others, (1) dropping records of data containing censored values and (2) assuming normality for censored observations and the residuals in the model. Across the methods considered, where researchers are interested primarily in the slope parameter, estimation bias is consistently reduced using the method of moments approach.     

Bootstrap confidence bands for the CDF using ranked-set sampling

In ranked-set sampling (RSS), a stratification by ranks is used to obtain a sample that tends to be more informative than a simple random sample of the same size. Previous work has shown that if the rankings are perfect, then one can use RSS to obtain Kolmogorov–Smirnov type confidence bands for the CDF that are narrower than those obtained under simple random sampling. Here we develop Kolmogorov–Smirnov type confidence bands that work well whether the rankings are perfect or not. These confidence bands are obtained by using a smoothed bootstrap procedure that takes advantage of special features of RSS. We show through a simulation study that the coverage probabilities are close to nominal even for samples with just two or three observations. A new algorithm allows us to avoid the bootstrap simulation step when sample sizes are relatively small.     

Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions

The main goal in this paper is to develop and apply stochastic simulation techniques for GARCH models with multivariate skewed distributions using the Bayesian approach. Both parameter estimation and model comparison are not trivial tasks and several approximate and computationally intensive methods (Markov chain Monte Carlo) will be used to this end. We consider a flexible class of multivariate distributions which can model both skewness and heavy tails. Also, we do not fix tail behaviour when dealing with fat tail distributions but leave it subject to inference.     

Nonparametric Bayesian inference for multivariate density functions using Feller priors

Multivariate density estimation plays an important role in investigating the mechanism of high-dimensional data. This article describes a nonparametric Bayesian approach to the estimation of multivariate densities. A general procedure is proposed for constructing Feller priors for multivariate densities and their theoretical properties as nonparametric priors are established. A blocked Gibbs sampling algorithm is devised to sample from the posterior of the multivariate density. A simulation study is conducted to evaluate the performance of the procedure.     

Global convergence of the log-concave MLE when the true distribution is geometric

Let X₁, …, X_n be i.i.d. from a discrete probability mass function (pmf) p. In Balabdaoui et al. [(2013), ‘Asymptotic Distribution of the Discrete Log-Concave mle and Some Applications’, JRSS-B, in press], the pointwise limit distribution of the log-concave maximum-likelihood estimator (MLE) was derived in both the well- and misspecified settings. In the well-specified setting, the geometric distribution was excluded, classified as being degenerate. In this article, we establish the global asymptotic theory of the log-concave MLE of a geometric pmf in all ?_q distances for q∈{1, 2, …}∪{∞}. We also show how these asymptotic results could be used in testing whether a pmf is geometric.     

Inverse gaussian accelerated test models based on cumulative damage

The failure of a system under environmental stress often can be described by an accelerated test model which incorporates the environmental variable L. Here, the failure of such a system at environmental level L is modeled as the first passage of accumulated damage to a critical threshold value. Assuming a discrete additive damage model leads to a Birnbaum–Saunders-type distribution for the failure time which can be closely approximated by an inverse Gaussian-type model. However, if a continuous damage model based on a Gaussian process is assumed, a more general family of inverse Gaussian accelerated test models is obtained. Three sets of failure data are discussed to illustrate the usefulness of this general family.     

Estimation of parameters for a Birnbaum–Saunders regression model with censored data

Little work has been published on the analysis of censored data for the Birnbaum–Saunders distribution (BISA). In this article, we implement the EM algorithm to fit a regression model with censored data when the failure times follow the BISA. Three approaches to implement the E-Step of the EM algorithm are considered. In two of these implementations, the M-Step is attained by an iterative least-squares procedure. The algorithm is exemplified with a single explanatory variable in the model.     

Bayesian inference under progressive type-I interval censoring

Bayesian estimation for population parameter under progressive type-I interval censoring is studied via Markov Chain Monte Carlo (MCMC) simulation. Two competitive statistical models, generalized exponential and Weibull distributions for modeling a real data set containing 112 patients with plasma cell myeloma, are studied for illustration. In model selection, a novel Bayesian procedure which involves a mixture model is proposed. Then the mix proportion is estimated through MCMC and used as the model selection criterion.     

A generalised Box–Cox transformation for the parametric estimation of clinical reference intervals

Parametric methods for the calculation of reference intervals in clinical studies often rely on the identification of a suitable transformation so that the transformed data can be assumed to be drawn from a Gaussian distribution. In this paper, the two-stage transformation recommended by the International Federation for Clinical Chemistry is compared with a novel generalised Box–Cox family of transformations. Investigation is also made of sample sizes needed to achieve certain criteria of reliability in the calculated reference interval. Simulations are used to show that the generalised Box–Cox family achieves a lower bias than the two-stage transformation. It was found that there is a possibility that the two-stage transformation will result in percentile estimates that cannot be back-transformed to obtain the required reference intervals, a difficulty not observed when using the generalised Box–Cox family introduced in this paper.     

On Pitman's measure of closeness of k-records

Pitman closeness of both the upper and lower k-record statistics to the population quantiles of a location–scale family of distributions is studied. For the population median, the Pitman-closest k-record is also determined. In the case of symmetric distributions, the Pitman closeness probabilities of k-record statistics are shown to be distribution-free, and explicit expressions are also derived for these probabilities. Exact expressions are derived for the required probabilities for uniform and exponential distributions. Numerical results are given for these families and also the Pitman-closest k-record is determined.