ABSTRACT
We have seen in the preceding chapters that a parameter can be estimated in various ways. This begs a natural question, how to compare the competing estimators. In the finite sample case, the estimators are compared by means of their risks, which are computed by reference to a suitable loss function. The choice of the loss function is a basic concept in this respect, but there is no unique choice of the loss which can be judged the most appropriate. The exact computation of a risk usually brings some technical difficulties, especially when the sample size is not so small. Fortunately, in an asymptotic setup, by allowing the sample size to increase indefinitely, it is often possible to induce various approximations and/or simplifications by which either the limit of a risk or the value of risk at the asymptotic distribution can be obtained in a closed form, what enables to compare the competing estimators asymptotically. The asymptotic normality considerations usually lead to the choice of squared error loss, and thus to the mean square error for a realvalued estimator and to quadratic risks in the multiparameter case. However, in the finite sample setup we can consider various loss criteria and their risk counterparts.
