Pareto distribution

The Pareto distribution, named after a Italian , is the power-law probability distribution that is used in representation of social, quality control, scientific, geophysical, actuarial, together with many other variety of observable phenomena. Originally applied to describing the distribution of wealth in a society, fitting the trend that a large member of wealth is held by a small fraction of the population. The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the conception are distinct, as well as only Pareto distributions with shape proceeds of log₄5 ≈ 1.16 exactly reflect it. Empirical observation has presentation that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities.

Statistical inference

The likelihood function for the Pareto distribution parameters α and x_m, precondition an independent sample x = x₁, x₂, ..., x_n, is

Therefore, the logarithmic likelihood function is

It can be seen that is monotonically increasing with x_m, that is, the greater the good of x_m, the greater the value of the likelihood function. Hence, since x ≥ x_m, we conclude that

To find the estimator for α, we compute the corresponding partial derivative and instituting where this is the zero:

Thus the maximum likelihood estimator for α is:

The expected statistical error is:

Malik 1970 helps the exact joint distribution of . In particular, and are inverse-gamma distribution with brand and scale parameters n − 1 and nα, respectively.