See https://en.wikipedia.org/wiki/Pareto_distribution for an introduction to the Pareto distribution.
Definition
Parameters (3): \(\xi\) (location), \(\alpha\) (scale), \(k\) (shape).
Range of \(x\): \(\xi < x \le \xi + \alpha / k\) if \(k>0\);
\(\xi \le x < \infty\) if \(k \le 0\).
Probability density function:
$$f(x) = \alpha^{-1} e^{-(1-k)y}$$
where \(y = -k^{-1}\log\{1 - k(x - \xi)/\alpha\}\) if \(k \ne 0\),
\(y = (x-\xi)/\alpha\) if \(k=0\).
Cumulative distribution function:
$$F(x) = 1-e^{-y}$$
Quantile function:
\(x(F) = \xi + \alpha[1-(1-F)^k]/k\) if \(k \ne 0\),
\(x(F) = \xi - \alpha \log(1-F)\) if \(k=0\).
\(k=0\) is the exponential distribution; \(k=1\) is the uniform distribution on the interval \(\xi < x \le \xi + \alpha\).
L-moments
L-moments are defined for \(k>-1\).
$$\lambda_1 = \xi + \alpha/(1+k)]$$
$$\lambda_2 = \alpha/[(1+k)(2+k)]$$
$$\tau_3 = (1-k)/(3+k)$$
$$\tau_4 = (1-k)(2-k)/[(3+k)(4+k)]$$
The relation between \(\tau_3\) and \(\tau_4\) is given by
$$\tau_4 = \frac{\tau_3 (1 + 5 \tau_3)}{5+\tau_3}$$
Parameters
If \(\xi\) is known, \(k=(\lambda_1 - \xi)/\lambda_2 - 2\) and \(\alpha=(1+k)(\lambda_1 - \xi)\);
if \(\xi\) is unknown, \(k=(1 - 3 \tau_3)/(1 + \tau_3)\), \(\alpha=(1+k)(2+k)\lambda_2\) and
\(\xi=\lambda_1 - (2+k)\lambda_2\).
Lmom.genpar
and par.genpar
accept input as vectors of equal length. In f.genpar
, F.genpar
, invF.genpar
and rand.genpar
parameters (xi
, alfa
, k
) must be atomic.