Modified profile likelihoods in models with stratum nuisance parameters

It is well known, at least through many examples, that when there are many nuisance parameters modified profile likelihoods often perform much better than the profile likelihood.Ordinary asymptotics almost totally fail to deal with this issue. For this reason, we study asymptotic properties of the profile and modified profile likelihoods in models for stratified data in a two-index asymptotics setting. This means that both the sample size of the strata, m, and the dimension of the nuisance parameter, q, may increase to infinity. It is shown that in this asymptotic setting modified profile likelihoods give improvements, with respect to the profile likelihood, in terms of consistency of estimators and of asymptotic distributional properties. In particular, the modified profile likelihood based statistics have the usual asymptotic distribution, provided that 1/m = o(q^-1/3), while the analogous condition for the profile likelihood is 1/m = o(q^-1).