Relative Efficiency of Higher Normed Estimators Over the Least Squares Estimator

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.