# Efficient Estimators and CRLB

(1 votes, average: 5.00 out of 5)

It has been reiterated that not all estimators are efficient. Even not all the MVUE are efficient. Then how do we quantify whether the estimator designed by us is efficient or not?

An efficient estimator is defined as the one that is
* Unbiased (Mean of the estimate = true value of the parameter)
* Attains CRLB

## How to Identify Efficient Estimators?

As mentioned in the previous article, the second partial derivative of log likelihood function of the observed signal model may be (not true always) written in a form like the one below.

$$\frac{\partial\;ln\;L(\mathbf{x};\theta) }{\partial \theta } = I(\theta)g(\mathbf{x}-\theta)$$

If we can write the CRLB equation in the above form, then the estimator $$g(\mathbf{x})$$ is an efficient estimator.

## Example:

In an another previous article, CRLB for an estimator that estimates the DC component from a set of observed samples (affected with AWGN noise) was derived. The intermediate step that derived the above requirement for the scenario is given below

From the above equation, it can be ascertained that the efficient estimator exists for the case and it is given by $$g(\mathbf{x})$$. The efficient estimator is simply given by sample mean of the observed samples.