This monograph is a collection of results recently obtained by the
authors. Most of these have been published, while others are awaitlng
publication. Our investigation has two main purposes. Firstly, we
discuss higher order asymptotic efficiency of estimators in regular
situa- tions. In these situations it is known that the maximum
likelihood estimator (MLE) is asymptotically efficient in some (not
always specified) sense. However, there exists here a whole class of
asymptotically efficient estimators which are thus asymptotically
equivalent to the MLE. It is required to make finer distinctions among
the estimators, by considering higher order terms in the expansions of
their asymptotic distributions. Secondly, we discuss asymptotically
efficient estimators in non- regular situations. These are situations
where the MLE or other estimators are not asymptotically normally
distributed, or where l 2 their order of convergence (or consistency) is
not n /, as in the regular cases. It is necessary to redefine the
concept of asympto- tic efficiency, together with the concept of the
maximum order of consistency. Under the new definition as asymptotically
efficient estimator may not always exist. We have not attempted to tell
the whole story in a systematic way. The field of asymptotic theory in
statistical estimation is relatively uncultivated. So, we have tried to
focus attention on such aspects of our recent results which throw light
on the area.