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.
We propose a novel, algebraic adjustment to the Wald statistic that can deliver significant improvements in inferential performance with only small 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 of 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. Ample analytical and numerical evidence and case-studies in prominent inferential settings, including logistic regression in the presence of nuisance parameters and analysis of multiple sclerosis lesions from MRI data, support our recommendation of adopting location-adjusted Wald statistics in statistical modelling.
Di Caterina, C. and Kosmidis, I. (2019). Location-adjusted Wald statistic for scalar parameters. Computational Statistics and Data Analysis, 138, 126-142