Quasi-likelihood ratio statistic for robust hypothesis testing in the presence of nuisance parameters

We discuss the problem of robust hypothesis testing about a scalar parameter of interest in the presence of a nuisance parameter. It is well-known that standard likelihood procedures are not robust with respect to model misspecification or the presence of outliers, which can badly affect hypothesis testing and model selection.Therefore, we discuss a quasi-profile loglikelihood with the standard distributional limit behaviour which, at the same time, assures robustness under small departures from the assumed model. This function is based on a profile estimating function, obtained by modifying a generalised profile score. A numerical study and an application about inference on the shape parameter of a gamma model, in the context of modelling personal-income distributions, are also considered.