Estimation of Superpopulation Parameters
199
can be plotted to examine plausible values of . A large-sample approximation to the 95% confi dence interval for  is obtainable from the likelihood ratio test as {: – 2 R max () # 3.84}.
When  is of higher dimension, gradient-based procedures such
as the quasi-Newton method or the conjugate gradient method can
be used for maximizing log L[ ˆ (), ]. As in the EM algorithm, these
methods will also increase the log-likelihood profi le at each iteration.
In principle, the EM algorithm could also be used to fi nd maximumlikelihood estimates for both and  simultaneously. However, the
part of the E-step that is relevant to  is diffi cult to carry out. It does not
appear to have a suffi ciently simple form for computations. We shall
not give the expressions here.
199
can be plotted to examine plausible values of . A large-sample approximation to the 95% confi dence interval for  is obtainable from the likelihood ratio test as {: – 2 R max () # 3.84}.
When  is of higher dimension, gradient-based procedures such
as the quasi-Newton method or the conjugate gradient method can
be used for maximizing log L[ ˆ (), ]. As in the EM algorithm, these
methods will also increase the log-likelihood profi le at each iteration.
In principle, the EM algorithm could also be used to fi nd maximumlikelihood estimates for both and  simultaneously. However, the
part of the E-step that is relevant to  is diffi cult to carry out. It does not
appear to have a suffi ciently simple form for computations. We shall
not give the expressions here.