Robust approximate Bayesian inference

We discuss an approach for deriving robust posterior distributions from $M$-estimating functions using Approximate Bayesian Computation (ABC) methods. In particular, we use $M$-estimating functions to construct suitable summary statistics in ABC algorithms. The theoretical properties of the robust posterior distributions are discussed. Special attention is given to the application of the method to linear mixed models. Simulation results and an application to a clinical study demonstrate the usefulness of the method. An R implementation is also provided in the robustBLME package.Comment: This is a revised and personal manuscript version of the article that has been accepted for publication by Journal of Statistical Planning and Inferenc

Approximate Bayesian Computation with composite score functions

Both Approximate Bayesian Computation (ABC) and composite likelihood methods are useful for Bayesian and frequentist inference, respectively, when the likelihood function is intractable. We propose to use composite likelihood score functions as summary statistics in ABC in order to obtain accurate approximations to the posterior distribution. This is motivated by the use of the score function of the full likelihood, and extended to general unbiased estimating functions in complex models. Moreover, we show that if the composite score is suitably standardised, the resulting ABC procedure is invariant to reparameterisations and automatically adjusts the curvature of the composite likelihood, and of the corresponding posterior distribution. The method is illustrated through examples with simulated data, and an application to modelling of spatial extreme rainfall data is discussed.Comment: Statistics and Computing (final version

A note on marginal posterior simulation via higher-order tail area approximations

We explore the use of higher-order tail area approximations for Bayesian simulation. These approximations give rise to an alternative simulation scheme to MCMC for Bayesian computation of marginal posterior distributions for a scalar parameter of interest, in the presence of nuisance parameters. Its advantage over MCMC methods is that samples are drawn independently with lower computational time and the implementation requires only standard maximum likelihood routines. The method is illustrated by a genetic linkage model, a normal regression with censored data and a logistic regression model

Minimum scoring rule inference

Proper scoring rules are methods for encouraging honest assessment of probability distributions. Just like likelihood, a proper scoring rule can be applied to supply an unbiased estimating equation for any statistical model, and the theory of such equations can be applied to understand the properties of the associated estimator. In this paper we develop some basic scoring rule estimation theory, and explore robustness and interval estimation properties by means of theory and simulations.Comment: 27 pages, 3 figure

Recent advances on Bayesian inference for P(X min Y )

We address the statistical problem of evaluating R = P(X < Y ), where X and Y are two independent random variables. Bayesian parametric inference about R, based on the marginal posterior density of R, has been widely discussed under various distributional assumptions on X and Y . This classical approach requires both elicitation of a prior on the complete parameter and numerical integration in order to derive the marginal distribution of R. In this paper, we discuss and apply recent advances in Bayesian inference based on higher-order asymptotics and on pseudo-likelihoods, and related matching priors, which allow to perform accurate inference on the parameter of interest only. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. The accuracy of the proposed methodology is illustrated both by numerical studies and by real-life data concerning clinical studie

Improved maximum likelihood estimation in heteroscedastic nonlinear regression models.

Nonlinear heteroscedastic regression models are a widely used class of models in applied statistics, with applications especially in biology, medicine or chemistry. Nonlinearity and variance heterogeneity can make likelihood estimation for a scalar parameter of interest rather inaccurate for small or moderate samples. In this paper, we suggest a new approach to point estimation based on estimating equations obtained from higher-order pivots for the parameter of interest. In particular, we take as an estimating function the modified directed likelihood. This is a higher-order pivotal quantity that can be easily computed in practice for nonlinear heteroscedastic models with normally distributed errors , using a recently developed S-PLUS library (HOA, 2000) . The estimators obtained from this procedure are a refinement of the maximum likelihood estimators, improving their small sample properties and keeping equivariance under reparameterisation. Two applications to real data sets are discussed

Accurate likelihood inference on the area under the ROC curve for small sample.

The accuracy of a diagnostic test with continuous-scale results is of high importance in clinical medicine. Receiver operating characteristics (ROC)curves, and in particular the area under the curve (AUC), are widely used to examine the effectiveness of diagnostic markers. Classical likelihood-based inference about the AUC has been widely studied under various parametric assumptions, but it is well-known that it can be inaccurate when the sample size is small, in particular in the presence of unknown parameters. The aim of this paper is to propose and discuss modern higher-order likelihood based procedures to obtain accurate point estimators and confidence intervals for the AUC. The accuracy of the proposed methodology is illustrated by simulation studies. Moreover, two real data examples are used to illustrate the application of the proposed methods

Robust inference in composite transformation models

The aim of this paper is to base robust inference about a shape parameter indexing a composite transformation model on a quasi- prole likelihood ratio test statistic. First, a general procedure is presented in order to construct a bounded prole estimating function for shape parameters. This method is based on a standard truncation argument from the theory of robustness. Hence, a quasi-likelihood test is derived. Numerical studies and applications to real data show that its use reveals extremely powerful, leading to improved inferences with respect to classical robust Wald and score-type test statistics

Robust prediction limits based on M-estimators

In this paper we discuss a robust solution to the problem of prediction. Following Barndorff-Nielsen and Cox (1996) and Vidoni (1998), we propose improved prediction limits based on M-estimators instead of maximum likelihood estimators. To compute these robust prediction limits, the expressions of the bias and variance of an M-estimator are required. Here a general asymptotic approximation for the bias of an M-estimator is derived. Moreover, by means of comparative studies in the context of affine transformation models, we show that the proposed robust procedure for prediction behaves in a similar manner to the classical one when the model is correctly specified, but it is designed to be stable in a neighborhood of the model

Quasi-profile loglikelihoods for unbiased estimating functions.

This paper presents a new quasi-profile loglikelihood with the standard kind of distributional limit behaviour, for inference about an arbitrary one-dimensional parameter of interest, based on unbiased estimating functions. The new function is obtained by requiring to the corresponding quasi-profile score function to have bias and information bias of order 0(1). We illustrate the use of the proposed pseudo-likelihood with an application for robust inference in linear models