# Relative Efficiency of Higher Normed Estimators Over the Least Squares Estimator

## TL;DR

This paper compares the efficiency of higher normed estimators, specifically $L_{2k}$, with the traditional least squares estimator, providing conditions, decision rules, and empirical validation for their relative performance.

## Contribution

It introduces a testable condition and decision rule for choosing between $L_{2k}$ and $L_2$ estimators, with a focus on the $L_4$ estimator and empirical validation.

## Key findings

- $L_{2k}$ estimator can be more efficient under certain conditions.
- A simple decision rule effectively guides estimator choice.
- Empirical results confirm the superiority of $L_{2k}$ in real data.

## Abstract

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.