Editing Maximum likelihood estimation (section)

=== Functional invariance ===
The maximum likelihood estimator selects the parameter value which gives the observed data the largest possible probability (or probability density, in the continuous case). If the parameter consists of a number of components, then we define their separate maximum likelihood estimators, as the corresponding component of the MLE of the complete parameter. Consistent with this, if <math>\widehat{\theta\,}</math> is the MLE for <math>\theta</math>, and if <math>g(\theta)</math> is any transformation of <math>\theta</math>, then the MLE for <math>\alpha=g(\theta)</math> is by definition<ref>{{cite book |first=Shelemyahu |last=Zacks |title=The Theory of Statistical Inference |location=New York |publisher=John Wiley & Sons |year=1971 |isbn=0-471-98103-6 |page=223 }}</ref>

<math display="block">\widehat{\alpha} = g(\,\widehat{\theta\,}\,). \,</math>

It maximizes the so-called [[Likelihood function#Profile likelihood|profile likelihood]]:

<math display="block">\bar{L}(\alpha) = \sup_{\theta: \alpha = g(\theta)} L(\theta). \, </math>

The MLE is also equivariant with respect to certain transformations of the data.  If <math>y=g(x)</math> where <math>g</math> is one to one and does not depend on the parameters to be estimated, then the density functions satisfy

<math display="block">f_Y(y) = f_X(g^{-1}(y)) \, |(g^{-1}(y))^{\prime}| </math>

and hence the likelihood functions for <math>X</math> and <math>Y</math> differ only by a factor that does not depend on the model parameters.

For example, the MLE parameters of the log-normal distribution are the same as those of the normal distribution fitted to the logarithm of the data. In fact, in the log-normal case if <math>X\sim\mathcal{N}(0, 1)</math>, then <math>Y=g(X)=e^{X} </math> follows a [[log-normal distribution]]. The density of Y follows with <math> f_X</math> standard [[Normal distribution|Normal]] and <math>g^{-1}(y) = \log(y) </math>, <math>|(g^{-1}(y))^{\prime}| = \frac{1}{y} </math> for <math> y > 0</math>.