Alessandra Salvan

Principles of statistical inference : from a neo-Fisherian perspective

In this book, an integrated introduction to statistical inference is provided from a frequentist likelihood-based viewpoint. Classical results are presented together with recent developments, largely built upon ideas due to R.A. Fisher. The term “neo-Fisherian” highlights this.After a unified review of background material (statistical models, likelihood, data and model reduction, first-order asymptotics) and inference in the presence of nuisance parameters (including pseudo-likelihoods), a self-contained introduction is given to exponential families, exponential dispersion models, generalized linear models, and group families. Finally, basic results of higher-order asymptotics are introduced (index notation, asymptotic expansions for statistics and distributions, and major applications to likelihood inference).The emphasis is more on general concepts and methods than on regularity conditions. Many examples are given for specific statistical models. Each chapter is supplemented with problems and bibliographic notes. This volume can serve as a textbook in intermediate-level undergraduate and postgraduate courses in statistical inference.

Bayesian Analysis in Regression Models Using Pseudo-Likelihoods

This article deals with the issue of using a suitable pseudo-likelihood, instead of an integrated likelihood, when performing Bayesian inference about a scalar parameter of interest in the presence of nuisance parameters. The proposed approach has the advantages of avoiding the elicitation on the nuisance parameters and the computation of multidimensional integrals. Moreover, it is particularly useful when it is difficult, or even impractical, to write the full likelihood function. We focus on Bayesian inference about a scalar regression coefficient in various regression models. First, in the context of non-normal regression-scale models, we give a theroetical result showing that there is no loss of information about the parameter of interest when using a posterior distribution derived from a pseudo-likelihood instead of the correct posterior distribution. Second, we present non trivial applications with high-dimensional, or even infinite-dimensional, nuisance parameters in the context of nonlinear normal heteroscedastic regression models, and of models for binary outcomes and count data, accounting also for possibile overdispersion. In all these situtations, we show that non Bayesian methods for eliminating nuisance parameters can be usefully incorporated into a one-parameter Bayesian analysis.

Point estimation based on confidence intervals: exponential families

Point estimators are usually judged in terms of their centering and spread properties, the most common measures being expectation and mean squared error. These measures, however, do not agree with the requirement of equivariance under reparameterization. In exponential families, when the parameter of interest is a scalar linear function of the natural parameter, an optimal equivariant estimator exists and is given by the zero-level optimal confidence interval. This optimal procedure is obtained by conditioning on the sufficient statistic for the nuisance parameter. However, the required conditional distribution is usually difficult to compute exactly. Quite accurate approximations are provided by recently developed higher-order asymptotic methods, in particular, by the modified directed likelihood introduced by Barndorff-Nielsen (1986). In this paper, an approximation for the optimal conditional estimator is obtained, using the modified directed likelihood as an estimating function. A simple explicit vers...

Modified Profile Likelihood for Fixed-Effects Panel Data Models

We show how modified profile likelihood methods, developed in the statistical literature, may be effectively applied to estimate the structural parameters of econometric models for panel data, with a remarkable reduction of bias with respect to ordinary likelihood methods. Initially, the implementation of these methods is illustrated for general models for panel data including individual-specific fixed effects and then, in more detail, for the truncated linear regression model and dynamic regression models for binary data formulated along with different specifications. Simulation studies show the good behavior of the inference based on the modified profile likelihood, even when compared to an ideal, although infeasible, procedure (in which the fixed effects are known) and also to alternative estimators existing in the econometric literature. The proposed estimation methods are implemented in an R package that we make available to the reader.

Improved inference on a scalar fixed effect of interest in nonlinear mixed-effects models

Likelihood-based inference on a scalar fixed effect of interest in nonlinear mixed-effects models usually relies on first-order approximations. If the sample size is small, tests and confidence intervals derived from first-order solutions can be inaccurate. An improved test statistic based on a modification of the signed likelihood ratio statistic is presented which was recently suggested by Skovgaard [1996. An explicit large-deviation approximation to one-parameter tests. Bernoulli 2, 145-165]. The finite sample behaviour of this statistic is investigated through a set of simulation studies. The results show that its finite-sample null distribution is better approximated by the standard normal than it is for its first-order counterpart. The R code used to run the simulations is freely available.

A note on directed adjusted profile likelihoods

Abstract Several adjustments to the profile likelihood have been proposed in recent years, to take into proper account the effects of fitting nuisance parameters. In some cases, adjusted profile likelihoods are higher-order approximations of suitable conditional or marginal target likelihoods. However, the xadjustments seem to provide accurate inference also when an exact marginal or conditional target likelihood is not available. Here, we consider adjusted profile likelihoods as approximations of a suitable general target likelihood. This is the likelihood for the parameter of interest with a known orthogonal nuisance parameter. Attention is focused on a scalar parameter of interest. Some new results are obtained concerning the null and non-null distributions of the directed likelihood calculated from an adjusted profile likelihood. In particular, we show that, while these distributions match the corresponding null and non-null distributions of the directed likelihood computed from the target likelihood up to order O(n−1/2) included, the agreement does not in general carry over to terms of order O(n−1), even if the information bias is of order O(n−1).

Planning sequential clinical trials: a review

Abstract Many recent developments in sequential analysis have been motivated by ethical requirements in the design of clinical trials together with the need of overcoming some difficulties of Walds theory. The purpose of this paper is to give a review of recently proposed methods for planning sequential clinical trials. Non-Bayesian and non-strictly-decision oriented methods for testing hypotheses concerning a normal mean with immediate response are considered. Methods are classified on the basis of the kind of comparison for which they were originally intended. This distinction has, in fact, an immediate reflection on different behaviours of the Average Sample Number functions. A general class of parametric boundaries is described, which yields many of the existing methods as particular cases.

Modified Profile Likelihood for Panel Data Models

We show how modified profile likelihood methods, developed in the statistical literature, may be effectively applied to estimate the structural parameters of econometric models for panel data, with a remarkable reduction of bias with respect to the ordinary likelihood methods. The implementation of these methods is illustrated in detail for certain static and dynamic models which are commonly used in economic applications. We consider, in particular, the truncated linear regression model, the first order autoregressive model, the (static and dynamic) logit model, and the (static and dynamic) probit model. Differently from static models, dynamic models include the lagged response variable among the regressors. For each of these models, we report the results of simulation studies showing the good behaviour of the proposed estimation methods, even with respect to an ideal, although infeasible, procedure. The methods are made available through an R package.

THE GEOMETRIC STRUCTURE OF THE EXPECTED/OBSERVED LIKELIHOOD EXPANSIONS*

Stochastic expansions of likelihood quantities are a basic tool for asymptotic inference. The traditional derivation is through ordinary Taylor expansions, rearranging terms according to their asymptotic order. The resulting expansions are called hereexpected/observed, being expressed in terms of the score vector, the expected information matrix, log likelihood derivatives and their joint moments. Though very convenient for many statistical purposes, expected/observed expansions are not usually written in tensorial form. Recently, within a differential geometric approach to asymptotic statistical calculations, invariant Taylor expansions based on likelihood yokes have been introduced. The resulting formulae are invariant, but the quantities involved are in some respects less convenient for statistical purposes. The aim of this paper is to show that, through an invariant Taylor expansion of the coordinates related to the expected likelihood yoke, expected/observed expansions up to the fourth asymptotic order may be re-obtained from invariant Taylor expansions. This derivation producesinvariant expected/observed expansions.

The Effects of Rounding on Likelihood Procedures

The aim of this paper is to investigate the robustness properties of likelihood inference with respect to rounding effects. Attention is focused on exponential families and on inference about a scalar parameter of interest, also in the presence of nuisance parameters. A summary value of the influence function of a given statistic, the local-shift sensitivity, is considered. It accounts for small fluctuations in the observations. The main result is that the local-shift sensitivity is bounded for the usual likelihood-based statistics, i.e. the directed likelihood, the Wald and score statistics. It is also bounded for the modified directed likelihood, which is a higher-order adjustment of the directed likelihood. The practical implication is that likelihood inference is expected to be robust with respect to rounding effects. Theoretical analysis is supplemented and confirmed by a number of Monte Carlo studies, performed to assess the coverage probabilities of confidence intervals based on likelihood procedures when data are rounded. In addition, simulations indicate that the directed likelihood is less sensitive to rounding effects than the Wald and score statistics. This provides another criterion for choosing among first-order equivalent likelihood procedures. The modified directed likelihood shows the same robustness as the directed likelihood, so that its gain in inferential accuracy does not come at the price of an increase in instability with respect to rounding.