The Pareto(IV) distribution, which is used in actuarial science,
economics, finance and telecommunications,
has a cumulative distribution function that can be written
$$F(y) = 1 - [1 + ((y-a)/b)^{1/g}]^{-s}$$
for \(y > a\), \(b>0\), \(g>0\) and \(s>0\).
The \(a\) is called the location parameter,
\(b\) the scale parameter,
\(g\) the inequality parameter, and
\(s\) the shape parameter.
The location parameter is assumed known otherwise the
Pareto(IV) distribution will not be a regular family.
This assumption is not too restrictive in modelling
because in typical applications this parameter is known,
e.g., in insurance and reinsurance it is pre-defined by
a contract and can be represented as a deductible or a
retention level.
The inequality parameter is so-called because of its
interpretation in the economics context. If we choose a
unit shape parameter value and a zero location parameter
value then the inequality parameter is the Gini index of
inequality, provided \(g \leq 1\).
The fitted values are currently the median, e.g.,
qparetoIV is used for paretoIV().
There are a number of special cases of the Pareto(IV) distribution.
These include the Pareto(I), Pareto(II), Pareto(III), and Burr family
of distributions.
Denoting \(PIV(a,b,g,s)\) as the Pareto(IV) distribution,
the Burr distribution \(Burr(b,g,s)\) is \(PIV(a=0,b,1/g,s)\),
the Pareto(III) distribution \(PIII(a,b,g)\) is \(PIV(a,b,g,s=1)\),
the Pareto(II) distribution \(PII(a,b,s)\) is \(PIV(a,b,g=1,s)\),
and
the Pareto(I) distribution \(PI(b,s)\) is \(PIV(b,b,g=1,s)\).
Thus the Burr distribution can be fitted using the
negloglink link
function and using the default location=0 argument.
The Pareto(I) distribution can be fitted using paretoff
but there is a slight change in notation: \(s=k\) and
\(b=\alpha\).