Editing Estimation theory (section)

====Cramér–Rao lower bound====
{{further|Cramér–Rao bound}}

To find the [[Cramér–Rao lower bound]] (CRLB) of the sample mean estimator, it is first necessary to find the [[Fisher information]] number
<math display="block">
\mathcal{I}(A)
=
\mathrm{E}
\left(
 \left[
  \frac{\partial}{\partial A} \ln p(\mathbf{x}; A)
 \right]^2
\right)
=
-\mathrm{E}
\left[
 \frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
\right]
</math>
and copying from above
<math display="block">
\frac{\partial}{\partial A} \ln p(\mathbf{x}; A)
=
\frac{1}{\sigma^2} \left[ \sum_{n=0}^{N-1}x[n] - N A \right]
</math>

Taking the second derivative
<math display="block">
\frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
=
\frac{1}{\sigma^2} (- N)
=
\frac{-N}{\sigma^2}
</math>
and finding the negative expected value is trivial since it is now a deterministic constant
<math>
-\mathrm{E}
\left[
 \frac{\partial^2}{\partial A^2} \ln p(\mathbf{x}; A)
\right]
=
\frac{N}{\sigma^2}
</math>

Finally, putting the Fisher information into
<math display="block">
\mathrm{var}\left( \hat{A} \right)
\geq
\frac{1}{\mathcal{I}}
</math>
results in
<math display="block">
\mathrm{var}\left( \hat{A} \right)
\geq
\frac{\sigma^2}{N}
</math>

Comparing this to the variance of the sample mean (determined previously) shows that the sample mean is ''equal to'' the Cramér–Rao lower bound for all values of <math>N</math> and <math>A</math>.
In other words, the sample mean is the (necessarily unique) [[efficient estimator]], and thus also the [[minimum variance unbiased estimator]] (MVUE), in addition to being the [[maximum likelihood]] estimator.