  In this article, we study the performance of the estimator that minimizes
$L_{2k}- $ order loss function (for $ k \ge \; 2 )$ against the estimators
which minimizes the $L_2-$ order loss function (or the least squares
estimator). Commonly occurring examples illustrate the differences in
efficiency between $L_{2k}$ and $L_2 -$ based estimators. We derive an
empirically testable condition under which the $L_{2k}$ estimator is more
efficient than the least squares estimator. We construct a simple decision rule
to choose between $L_{2k}$ and $L_2$ estimator. Special emphasis is provided to
study $L_{4}$ estimator. A detailed simulation study verifies the effectiveness
of this decision rule. Also, the superiority of the $L_{2k}$ estimator is
demonstrated in a real life data set.
