Efficiency of projected score methods in rectangular array asymptotics

The paper considers a rectangular array asymptotic embedding for multistratum data sets, in which both the number of strata and the number of within–stratum replications increase, and at the same rate. It is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non–zero mean in its asymptotic normal distribution. By using a projection operator on the score function, an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the Cramér–Rao lower bound. The adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator.