# Applying Cramer Rao Lower Bound (CRLB) to find a Minimum Variance Unbiased Estimator (MVUE)

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It was mentioned in one of the earlier articles that CRLB may provide a way to find a MVUE (Minimum Variance Unbiased Estimators).

## Theorem:

There exists an unbiased estimator that attains CRLB if and only if,

Here $$ln \; L(\mathbf{x};\theta)$$ is the log likelihood function of x parameterized by $$\theta$$ – the parameter to be estimated, $$I(\theta)$$ is the Fisher Information and $$g(x)$$ is some function.

Then, the estimator that attains CRLB is given by

## Steps to find MVUE using CRLB:

If we could write the equation (as given above) in terms of Fisher Matrix and some function $$g(x)$$ then $$g(x)$$ is a Minimum Variable Unbiased Estimator.
1) Given a signal model $$x$$, compute $$\frac{\partial\;ln\;L(\mathbf{x};\theta) }{\partial \theta }$$
2) Check if the above computation can be put in the form like the one given in the above theorem
3) Then $$g(\mathbf{x})$$ given an MVUE

Let’s look at how CRLB can be used to find an MVUE for a signal that has a DC component embedded in AWGN noise.

## Finding a MVUE to estimate DC component embedded in noise:

Consider the signal model where a DC component – $$A$$ is embedded in an AWGN noise with zero mean and variance=$$\sigma$$.
Our goal is to find an MVUE that could estimate the DC component from the observed samples $$x[n]$$.

$$x[n] = A + w[n], \;\;\; n=0,1,2,\cdots,N-1$$

We calculate CRLB and see if it can help us find a MVUE.

From the previous derivation

From the above equation we can readily identify $$I(A)$$ and $$g(\mathbf{x})$$ as follows

Thus,the Fisher Information $$I(A)$$ and the MVUE $$g(\mathbf{x})$$ are given by

Thus for a signal model which has a DC component in AWGN, the sample mean of observed samples $$x[n]$$ gives a Minimum Variance Unbiased Estimator to estimate the DC component.