Is unbiasedness a necessary condition for an estimator to be efficient?
For example, if $hat {theta}= frac{sum_i^n X_i}{3}$, I assume $hat {theta}$ can't be efficient in a Cramer-Rao lower bound context because $E[hat {theta}]= frac {theta}{3}$.
Contents
hide
Best Answer
Clearly not.
A possible way to compare two estimators is to use Mean Squared Error : $begin{align*} MSE = Bias^2 + Variance end{align*}$.
There are some biased estimators with very good variances, this being better choices than some other unbiased estimators with awfullly high variances.
See this blog post for an illustration in Python.
Similar Posts:
- Solved – Cramer-Rao lower bound in a Gamma distribution
- Solved – Intuitive explanation of desirable properties (Unbiasedness, Consistency, Efficiency) of statistical estimators
- Solved – Does efficiency imply unbiased and consistency
- Solved – Sufficient statistics, MLE and unbiased estimators of uniform type distribution
- Solved – When can’t Cramer-Rao lower bound be reached