*! version 1.0.0 29jul2010

help paretowmlMichal Brzezinski (July 2010)

Title

paretowml -- Estimate the shape parameter a for
the Pareto (Type I) distribution using weighted maximum likelihood estimator

Syntax

paretowml [varname] [, options]

k(scalar) the maximum number of extreme order statistics used in estimation

x0(scalar) the value of scale parameter for Pareto model

c(scalar) robustness constant

If neither k nor x0 options are used, x0 is set to the minimum value of varname.

Description

paretowml estimates the shape parameter a for
the Pareto (Type I) distribution using weighted maximum likelihood estimator (WMLE). As it
is well-known ML estimation of a (see paretofit)
is very sensitive to extreme observations. WMLE downweights observations, which are inconsistent
with Pareto model in terms of the residuals with respect to the Pareto regression model.
For details, see (Dupuis and Victoria-Feser, 2006).

For implementations of non-robust and robust criteria for choosing
scale parameterx0, introduced in Dupuis and Victoria-Feser (2006), see nrpec and rpec, if installed.

Examples

To estimate a with x0 = 1000:

. paretowml varname, x0(1000)

To estimate a using x0 estimated by
applying robust Pareto error criterion of Dupuis and Victoria-Feser (2006):

. rpec varname, k(100)

. scalar x0=r(x0)

. paretowml varname, x0(x0)

References

Dupuis, Debbie J. and Victoria-Feser, Maria-Pia (2006). A Robust Prediction
Error Criterion for Pareto Modelling of Upper Tails. The Canadian Journal of Statistics, 34 (4), 639-658.

Also see

Online: help for nrpec, rpec, paretofit, paretoobre, hillp if installed.

Author

Michal Brzezinski <mbrzezinski@wne.uw.edu.pl>, Faculty of Economic Sciences, University of Warsaw, Poland.