We concern ourselves with the methods for testing the overall goodness of fit of a parametric family of link functions used for modeling the conditional mean of the response variable Y given the covariates X = x ∈ R P. The null hypothesis is that the conditional mean function is a known functional depending on βx and a finite number of parameters θ = (θ1,…, θq), where β is a p-dimensional row vector of regression parameters and x is a column vector. The proposed test statistic is derived from an “information” equivalence result and a dimension-reduction technique. The new test is very simple in computation. Also, it is generally consistent against broad class of alternatives and, asymptotically, the null distribution is independent of the underlying distribution of Y, given X = x. Practical examples are given to show the advantage of the proposed test. Furthermore, power comparisons with the test used by Su and Wei are also performed to indicate the usefulness of the new test. Particularly, we find that the new test has good power performance in discriminating between the probit and logit links.
Journal of the American Statistical Association 89(426), pp.657-664