Editing Weibull distribution (section)

===Maximum likelihood===
The [[Maximum likelihood estimation|maximum likelihood estimator]] for the <math>\lambda</math> parameter given <math>k</math> is<ref name="Cohen1965"/>

:<math>\widehat \lambda = \left(\frac{1}{n} \sum_{i=1}^n x_i^k \right)^\frac{1}{k} </math>

The maximum likelihood estimator for <math>k</math> is the solution for ''k'' of the following equation<ref name="Sornette, D. 2004">{{cite book
 | author = Sornette, D.
 | year = 2004
 | title = Critical Phenomena in Natural Science: Chaos, Fractals, Self-organization, and Disorder}}.</ref>
:<math>
  0 =  \frac{\sum_{i=1}^n x_i^k \ln x_i }{\sum_{i=1}^n x_i^k }
                  - \frac{1}{k} - \frac{1}{n} \sum_{i=1}^n \ln x_i
</math>

This equation defines <math>\widehat k</math> only implicitly, one must generally solve for <math>k</math> by numerical means.

When <math>x_1 > x_2 > \cdots > x_N</math> are the <math>N</math> largest observed samples from a dataset of more than <math>N</math> samples, then the maximum likelihood estimator for the <math>\lambda</math> parameter given <math>k</math> is<ref name="Sornette, D. 2004"/>

:<math>\widehat \lambda^k = \frac{1}{N} \sum_{i=1}^N (x_i^k - x_N^k)</math>

Also given that condition, the maximum likelihood estimator for <math>k</math> is{{citation needed|date=December 2017}}
:<math>
  0 = \frac{\sum_{i=1}^N (x_i^k \ln x_i -  x_N^k \ln x_N)}
                       {\sum_{i=1}^N (x_i^k - x_N^k)}
                  - \frac{1}{N} \sum_{i=1}^N \ln x_i
</math>

Again, this being an implicit function, one must generally solve for <math>k</math> by numerical means.