Modifications to the profile likelihood in models with incidental nuisance parameters.

We study asymptotic properties of the profile and modified profile likelihoods in models for clustered data with incidental nuisance parameters. To his end , we use a two-index asymptotics setting. This means that both the sample size of the clusters, m, and the dimension of the nuisance parameter, q, may increase to infinity. It is shown that modified profile likelihoods give improvements, with respect to the profile likelihood, on consistency of estimates and on asymptotic distribution properties. In particular, the profile likelihood based statistics have the usual asymptotic distribution, provided that 1/m=o(q^(-1)) , while the analogous condition for modified profile likelihoods in 1/m=o(q^(-1/3)).