hoa: An R package bundle for higher order likelihood inference

The likelihood function represents the basic ingredient of many commonly used statistical methods for estimation, testing and the calculation of confidence intervals. In practice, much application of likelihood inference relies on first order asymptotic results such as the central limit theorem. The approximations can, however, be rather poor if the sample size is small or, generally, when the average information available per parameter is limited. Thanks to the great progress made over the past twenty-five years or so in the theory of likelihood inference, very accurate approximations to the distribution of statistics such as the likelihood ratio have been developed. These not only provide modifications to well-established approaches, which result in more accurate inferences, but also give insight on when to rely upon first order methods. We refer to these developments as higher order asymptotics. One intriguing feature of the theory of higher order likelihood asymptotics is that relatively simple and familiar quantities play an essential role. The higher order approximations discussed in this paper are for the significance function, which we will use to set confidence limits or to calculate the p-value associated with a particular hypothesis of interest. We start with a concise overview of the approximations used in the remainder of the paper. Our first example is an elementary one-parameter model where one can perform the calculations easily, chosen to illustrate the potential accuracy of the procedures. Two more elaborate examples, an application of binary logistic regression and a nonlinear growth curve model, follow. All examples are carried out using the R code of the hoa package bundle.