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

Relative Efficiency of Higher Normed Estimators Over the Least Squares Estimator