Luigi Pace

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.

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).

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.

Tensors and likelihood expansions in the presence of nuisance parameters

Stochastic expansions of likelihood quantities are usually derived through oridinary Taylor expansions, rearranging terms according to their asymptotic order. The most convenient form for such expansions involves the score function, the expected information, higher order log-likelihood derivatives and their expectations. Expansions of this form are called expected/observed. If the quantity expanded is invariant or, more generally, a tensor under reparameterisations, the entire contribution of a given asymptotic order to the expected/observed expansion will follow the same transformation law. When there are no nuisance parameters, explicit representations through appropriate tensors are available. In this paper, we analyse the geometric structure of expected/observed likelihood expansions when nuisance parameters are present. We outline the derivation of likelihood quantities which behave as tensors under interest-respectign reparameterisations. This allows us to write the usual stochastic expansions of profile likelihood quantities in an explicitly tensorial form.

Remedying the Neyman–Scott phenomenon in model discrimination

The objective of this paper is to investigate through simulation the possible presence of the incidental parameters problem when performing frequentist model discrimination with stratified data. In this context, model discrimination amounts to considering a structural parameter taking values in a finite space, with k points, k≥2. This setting seems to have not yet been considered in the literature about the Neyman–Scott phenomenon. Here we provide Monte Carlo evidence of the severity of the incidental parameters problem also in the model discrimination setting and propose a remedy for a special class of models. In particular, we focus on models that are scale families in each stratum. We consider traditional model selection procedures, such as the Akaike and Takeuchi information criteria, together with the best frequentist selection procedure based on maximization of the marginal likelihood induced by the maximal invariant, or of its Laplace approximation. Results of two Monte Carlo experiments indicate that when the sample size in each stratum is fixed and the number of strata increases, correct selection probabilities for traditional model selection criteria may approach zero, unlike what happens for model discrimination based on exact or approximate marginal likelihoods. Finally, two examples with real data sets are given.