Restricted minimax robust designs for misspecified regression models

The authors propose and explore new regression designs. Within a particular parametric class, these designs are minimax robust against bias caused by model misspecification while attaining reasonable levels of efficiency as well. The introduction of this restricted class of designs is motivated by a desire to avoid the mathematical and numerical intractability found in the unrestricted minimax theory. Robustness is provided against a family of model departures sufficiently broad that the minimax design measures are necessarily absolutely continuous. Examples of implementation involve approximate polynomial and second order multiple regression.     

Nonparametric smooth estimation of the expected inactivity time function

The expected inactivity time (EIT) function (also known as the mean past lifetime function) is a well known reliability function which has application in many disciplines such as survival analysis, actuarial studies and forensic science, to name but a few. In this paper, we use a fixed design local polynomial fitting technique to obtain estimators for the EIT function when the lifetime random variable has an unknown distribution. It will be shown that the proposed estimators are asymptotically unbiased, consistent and also, when standardized, has an asymptotic normal distribution. An optimal bandwidth, which minimizes the AMISE (asymptotic mean integrated squared error) of the estimator, is derived. Numerical examples based on simulated samples from various lifetime distributions common in reliability studies will be presented to evaluate the performances of these estimators. Finally, three real life applications will also be presented to further illustrate the wide applicability of these estimators.     

Bayesian influence diagnostics in radiocarbon dating

Linear models constitute the primary statistical technique for any experimental science. A major topic in this area is the detection of influential subsets of data, that is, of observations that are influential in terms of their effect on the estimation of parameters in linear regression or of the total population parameters. Numerous studies exist on radiocarbon dating which propose a value consensus and remove possible outliers after the corresponding testing. An influence analysis for the value consensus from a Bayesian perspective is developed in this article.     

Lower posterior death probabilities from a quick medical response in road traffic accidents

Introduction: We use data from Spain on roads and motorways traffic accidents in May 2004 to quantify the statistical association between quick medical response time and mortality rate. Method: Probit and logit parameters are estimated by a Bayesian method in which samples from the posterior densities are obtained through an MCMC simulation scheme. We provide posterior credible intervals and posterior partial effects of a quick medical response at several time levels over which we express our prior beliefs. Results: A reduction of 5 min, from a 25-min response-time level, is associated with lower posterior probabilities of death in roads and motorways accidents of 24% and 30%, respectively.     

Parameter estimation of nonlinear mixed-effects models using first-order conditional linearization and the EM algorithm

Nonlinear mixed-effects (NLME) models are flexible enough to handle repeated-measures data from various disciplines. In this article, we propose both maximum-likelihood and restricted maximum-likelihood estimations of NLME models using first-order conditional expansion (FOCE) and the expectation–maximization (EM) algorithm. The FOCE-EM algorithm implemented in the ForStat procedure SNLME is compared with the Lindstrom and Bates (LB) algorithm implemented in both the SAS macro NLINMIX and the S-Plus/R function nlme in terms of computational efficiency and statistical properties. Two realworld data sets an orange tree data set and a Chinese fir (Cunninghamia lanceolata) data set, and a simulated data set were used for evaluation. FOCE-EM converged for all mixed models derived from the base model in the two realworld cases, while LB did not, especially for the models in which random effects are simultaneously considered in several parameters to account for between-subject variation. However, both algorithms had identical estimated parameters and fit statistics for the converged models. We therefore recommend using FOCE-EM in NLME models, particularly when convergence is a concern in model selection.     

Modification of Tukey's additivity test

In this paper we discuss testing for an interaction in the two-way ANOVA with just one observation per cell. The known results are reviewed and a simulation study is performed to evaluate type I and type II risks of the tests. It is shown that the Tukey and Mandel additivity tests have very low power in case of more general interaction scheme. A modification of Tukey's test is developed to resolve this issue. All tests mentioned in the paper have been implemented in R package Additivity Tests.     

Robust model-based sampling designs

We investigate methods for the design of sample surveys, and address the traditional resistance of survey samplers to the use of model-based methods by incorporating model robustness at the design stage. The designs are intended to be sufficiently flexible and robust that resulting estimates, based on the designer’s best guess at an appropriate model, remain reasonably accurate in a neighbourhood of this central model. Thus, consider a finite population of N units in which a survey variable Y is related to a q dimensional auxiliary variable x. We assume that the values of x are known for all N population units, and that we will select a sample of n≤N population units and then observe the n corresponding values of Y. The objective is to predict the population total $T=\sum_{i=1}^{N}Y_{i}$ . The design problem which we consider is to specify a selection rule, using only the values of the auxiliary variable, to select the n units for the sample so that the predictor has optimal robustness properties. We suppose that T will be predicted by methods based on a linear relationship between Y—possibly transformed—and given functions of x. We maximise the mean squared error of the prediction of T over realistic neighbourhoods of the fitted linear relationship, and of the assumed variance and correlation structures. This maximised mean squared error is then minimised over the class of possible samples, yielding an optimally robust (‘minimax’) design. To carry out the minimisation step we introduce a genetic algorithm and discuss its tuning for maximal efficiency.     

Iterative bias reduction: a comparative study

Multivariate nonparametric smoothers, such as kernel based smoothers and thin plate splines smoothers, are adversely impacted by the sparseness of data in high dimension, also known as the curse of dimensionality. Adaptive smoothers, that can exploit the underlying smoothness of the regression function, may partially mitigate this effect. This paper presents a comparative simulation study of a novel adaptive smoother (IBR) with competing multivariate smoothers available as package or function within the R language and environment for statistical computing. Comparison between the methods are made on simulated datasets of moderate size, from 50 to 200 observations, with two, five or 10 potential explanatory variables, and on a real dataset. The results show that the good asymptotic properties of IBR are complemented by a very good behavior on moderate sized datasets, results which are similar to those obtained with Duchon low rank splines.     

Nonparametric comparison of curves with dependent errors

Suppose that data {(x _l,i,n , y _l,i,n ): l?=?1, …, k; i?=?1, …, n} are observed from the regression models: Y _l,i,n ?=?m _l (x _l,i,n )?+?? _l,i,n , l?=?1, …, k, where the regression functions {m _l } _l=1 ^k are unknown and the random errors {? _l,i,n } are dependent, following an MA(∞) structure. A new test is proposed for testing the hypothesis H ₀: m ₁?=?·?·?·?=?m _k , without assuming that {m _l } _l=1 ^k are in a parametric family. The criterion of the test derives from a Crámer-von-Mises-type functional based on different distances between {[mcirc]} _l and {[mcirc]} _s , l?≠?s, l, s?=?1, …, k, where {[mcirc] _l } _l=1 ^k are nonparametric Gasser–Müller estimators of {m _l } _l=1 ^k . A generalization of the test to the case of unequal design points, with different sample sizes {n _l } _l=1 ^k and different design densities {f _l } _l=1 ^k , is also considered. The asymptotic normality of the test statistic is obtained under general conditions. Finally, a simulation study and an analysis with real data show a good behavior of the proposed test.