Inference about a scalar parameter of interest is a core statistical task that has attracted immense research in statistics. The Wald statistic is a prime candidate for the task, on the grounds of the asymptotic validity of the standard normal approximation to its finite-sample distribution, simplicity and low computational cost. It is well known, though, that this normal approximation can be inadequate, especially when the sample size is small or moderate relative to the number of parameters. A novel, algebraic adjustment to the Wald statistic is proposed, delivering significant improvements in inferential performance with only small implementation and computational overhead, predominantly due to additional matrix multiplications. The Wald statistic is viewed as an estimate of a transformation of the model parameters and is appropriately adjusted, using either maximum likelihood or reduced-bias estimators, bringing its expectation asymptotically closer to zero. The location adjustment depends on the expected information, an approximation to the bias of the estimator, and the derivatives of the transformation, which are all either readily available or easily obtainable in standard software for a wealth of models. An algorithm for the implementation of the location-adjusted Wald statistics in general models is provided, as well as a bootstrap scheme for the further scale correction of the location-adjusted statistic. Ample analytical and numerical evidence is presented for the adoption of the location-adjusted statistic in prominent modelling settings, including inference about log-odds and binomial proportions, logistic regression in the presence of nuisance parameters, beta regression, and gamma regression. The location-adjusted Wald statistics are used for the construction of significance maps for the analysis of multiple sclerosis lesions from MRI data.

Directional tests to compare incomplete undirected graphs are developed in the general context of covariance selection for Gaussian graphical models. The exactness of the underlying saddlepoint approximation is proved for chordal graphs and leads to exact control of the size of the tests, given that the only approximation error involved is due to the numerical calculation of two scalar integrals. Although exactness is not guaranteed for non-chordal graphs, the ability of the saddlepoint approximation to control the relative error leads the directional test to overperform its competitors even in these cases. The accuracy of our proposal is verified by simulation experiments under challenging scenarios, where inference via standard asymptotic approximations to the likelihood ratio test and some of its higher-order modifications fails. The directional approach is used to illustrate the assessment of Markovian dependencies in a dataset from a veterinary trial on cattle. A second example with microarray data shows how to select the graph structure related to genetic anomalies due to acute lymphocytic leukemia.

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Composite likelihood has shown promise in settings where the number of parameters p is large due to its ability to break down complex models into simpler components, thus enabling inference even when the full likelihood is not tractable. Although there are a number of ways to formulate a valid composite likelihood in the finite-p setting, there does not seem to exist agreement on how to construct composite likelihoods that are comp utationally efficient and statistically sound when p is allowed to diverge. This article introduces a method to select sparse composite likelihoods by minimizing a criterion representing the statistical efficiency of the implied estimator plus an L 1 -penalty discouraging the inclusion of too many sub-likelihood terms. Conditions under which consistent model selection occurs are studied. Examples illustrating the procedure are analysed in detail and applied to real data.