Editing Pareto distribution (section)

==Statistical inference==

===Estimation of parameters===
The [[likelihood function]] for the Pareto distribution parameters ''α'' and ''x''<sub>m</sub>, given an independent [[sample (statistics)|sample]] ''x'' =&nbsp;(''x''<sub>1</sub>,&nbsp;''x''<sub>2</sub>,&nbsp;...,&nbsp;''x<sub>n</sub>''), is

: <math>L(\alpha, x_\mathrm{m}) = \prod_{i=1}^n \alpha \frac {x_\mathrm{m}^\alpha} {x_i^{\alpha+1}} = \alpha^n x_\mathrm{m}^{n\alpha} \prod_{i=1}^n \frac {1}{x_i^{\alpha+1}}.</math>

Therefore, the logarithmic likelihood function is

: <math>\ell(\alpha, x_\mathrm{m}) = n \ln \alpha + n\alpha \ln x_\mathrm{m} - (\alpha + 1) \sum_{i=1} ^n \ln x_i.</math>

It can be seen that <math>\ell(\alpha, x_\mathrm{m})</math> is monotonically increasing with ''x''<sub>m</sub>, that is, the greater the value of ''x''<sub>m</sub>, the greater the value of the likelihood function. Hence, since ''x'' ≥ ''x''<sub>m</sub>, we conclude that

: <math>\widehat x_\mathrm{m} = \min_i {x_i}.</math>

To find the [[estimator]] for ''α'', we compute the corresponding partial derivative and determine where it is zero:

: <math>\frac{\partial \ell}{\partial \alpha} = \frac{n}{\alpha} + n \ln x_\mathrm{m} - \sum _{i=1}^n \ln x_i = 0.</math>

Thus the [[maximum likelihood]] estimator for ''α'' is:

: <math>\widehat \alpha = \frac{n}{\sum _i  \ln (x_i/\widehat x_\mathrm{m}) }.</math>

The expected statistical error is:<ref>{{cite journal |author=M. E. J. Newman |year=2005 |title=Power laws, Pareto distributions and Zipf's law |journal=[[Contemporary Physics]] |volume=46 |issue=5 |pages=323–51| arxiv=cond-mat/0412004 |doi=10.1080/00107510500052444 |bibcode=2005ConPh..46..323N|s2cid=202719165 }}</ref>

: <math>\sigma = \frac {\widehat \alpha} {\sqrt n}. </math>

Malik (1970)<ref>{{cite journal |author=H. J. Malik |year=1970 |title=Estimation of the Parameters of the Pareto Distribution |journal=Metrika |volume=15|pages=126–132 |doi=10.1007/BF02613565 |s2cid=124007966 }}</ref> gives the exact joint distribution of <math>(\hat{x}_\mathrm{m},\hat\alpha)</math>. In particular, <math>\hat{x}_\mathrm{m}</math> and <math>\hat\alpha</math> are [[Independence (probability theory)|independent]] and <math>\hat{x}_\mathrm{m}</math> is Pareto with scale parameter ''x''<sub>m</sub> and shape parameter ''nα'', whereas <math>\hat\alpha</math> has an [[inverse-gamma distribution]] with shape and scale parameters ''n''&nbsp;−&nbsp;1 and ''nα'', respectively.