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. Copyright 2003 Royal Statistical Society.