Examples where the Maximum Likelihood Estimator has Small Asymptotic Efficiency


Ram Dahiya                                        S. Blumenthal  
Dept. of Mathematics and Statistics     and       Dept. of Statistics
Old Dominion University                           The Ohio State University



                        ABSTRACT

The maximum likelihood estimators (MLE) under some general conditions,
are asymptotically efficient and normally distributed. Since most text
books avoid discussing the asymptotic distribution of the MLE in the
non-regular case when support of the distribution depends on the
parameter, students tend to be under the misimpression that the MLE is
always asymptotically efficient. In this paper, we give two examples
where the MLE is asymptotically only 50% efficient, and one where it
is only 33% efficient. Another interesting feature of the first two
examples is that the asymptotic distribution of the MLE is exponential
instead of the usual normal distribution for the regular case and is
not centered at the parameter being estimated.