Fitting a Pareto (Type I) distribution by ML to unit record data
paretofit var [weight] [if exp] [in range] [, avar(varlist) cdf(cdfname) pdf(pdfname) robust cluster(varname) from(#) level(#) maximize_options ]
by and svy prefixes are allowed (but not jointly); see prefix.
fweights, aweights, pweights, and iweights are allowed; see weight.
Description
paretofit fits by ML a Pareto (Type I) distribution to sample observations on a random variable var. Unit record data are assumed (rather than grouped data).
The Pareto distribution is named after the Italian economist Vilfredo Pareto (1848-1923). It is one of the most famous and widely studied statistical size distributions. It is well-known for approximating wealth distributions, but has applications in many different fields (e.g. for size distibutions of human setllements, sand particles, word frequencies, or for assessing portfolio risk). See Kleiber and Kotz (2003) for a comprehensive review of the Pareto (and other) distributions.
The likelihood function for a sample of observations on var is specified as the product of the densities for each observation (weighted where relevant), and is maximized using ml model lf. A closed-form expression for the ML estimator of the Pareto Type I shape parameter is readily available but estimation with ml model allows us to accommodate various sample design easily, as well as inclusion of covariates.
Options
x0(scalar) specifies the scale parameter of the Pareto distribution (see formula below). The Pareto distribution is fitted only to sample observations where var>=x0. By default, x0 is set to the minimum value of var (within the sub-sample identified by if and in clauses).
avar(varlist) allows the user to specify the shape parameter of the distribution as a function of the covariates specified in varlist. A constant term is always included.
stats displays selected distributional statistics implied by the Pareto parameter estimate: quantiles, cumulative shares of total var at quantiles (i.e. the Lorenz curve ordinates), the mode, mean, standard deviation, variance, half the coefficient of variation squared, Gini coefficient, and quantile ratios p90/p10, p75/p25. This option is not available together with avar(varlist).
poorfrac(#) displays the estimated proportion with values of var less than the cut-off specified by #. This option may be specified when replaying results.
cdf(cdfname) creates a new variable cdfname containing the estimated Pareto c.d.f. value F(x) for each x.
pdf(pdfname) creates a new variable pdfname containing the estimated Pareto p.d.f. value f(x) for each x.
robust specifies that the Huber/White/sandwich estimator of variance is to be used in place of the traditional calculation; see [U] 23.14 Obtaining robust variance estimates. robust combined with cluster() allows observations which are not independent within cluster (although they must be independent between clusters). pweights imply robust.
cluster(varname) specifies that the observations are independent across groups (clusters) but not necessarily within groups. varname specifies to which group each observation belongs; e.g., cluster(personid) in data with repeated observations on individuals. See [U] 23.14 Obtaining robust variance estimates. cluster() can be used with pweights to produce estimates for unstratified cluster-sampled data. Use the svy prefix for full complex survey design support. Specifying cluster() implies robust.
from(#) specifies a starting value for the maximum likelhood estimation.
level(#) specifies the confidence level, in percent, for the confidence intervals of the coefficients; see help level.
nolog suppresses the iteration log.
maximize_options control the maximization process. The options available are those shown by maximize. If you are seeing many "(not concave)" messages in the iteration log, using the difficult or technique options may help convergence.
Saved results
In addition to the usual results saved after ml, paretofit saves the following, if no covariates have been specified and the relevant options are used:
e(ba) is the estimated Pareto Type I shape parameter.
e(cdfvar) and e(pdfvar) are the variable names specified for the c.d.f. and the p.d.f.
e(mode), e(mean), e(var), e(sd), e(i2), and e(gini) are the estimated mode, mean, variance, standard deviation, half coefficient of variation squared, Gini coefficient. e(pX), and e(LpX) are the quantiles, and Lorenz ordinates, where X = {1, 5, 10, 20, 25, 30, 40, 50, 60, 70, 75, 80, 90, 95, 99}.
Formulae
The Pareto (Type I) distribution has cumulative distribution function (c.d.f.)
F(x) = 1 - { x0 / x }^a
where a>0 is a shape parameter (estimated by paretofit), x0 is a scale parameter, and x >= x0 > 0 is a random variable. The right tail of a Pareto distribution is heavier as a is smaller.
The probability density function (p.d.f.) is
f(x) = a*(x0^a) / x^(a+1).
The formulae used to derive the distributional summary statistics presented (optionally) are as follows. The r-th moment about the origin is given by
a*(x0^r) / (a-r)
which exists only if r<a (Kleiber and Kotz, 2003, p. 70). It follows that
mean = a*x0 / (a-1)
variance = a*(x0^2) / [ a*(a-2)*(a-1)^2 ]
from which the standard deviation and half the squared coefficient of variation can be derived. These three statistics are defined only where a>2. The density is decreasing, so the mode is simply
mode = x0.
The quantiles are derived by inverting the distribution function:
x_s = x0*(1-s)^(-1/a), for each 0 < s = F(x_s) < 1.
The median is therefore
median = x0*(2^(1/a)).
The Gini coefficient of inequality is given by
Gini = 1 / (2a - 1).
The Lorenz curve ordinates at each s = F(x_s) are given by
L(s) = 1 - (1 - s)^{1 - 1/a).
Examples
. paretofit x
. paretofit x [fw=wgt]
. paretofit
. paretofit x [aw=wgt] , x0(20)
. paretofit, stats poorfrac(100) x0(50)
. paretofit x, avar(age sex) x0(50)
Authors
Stephen P. Jenkins <stephenj@essex.ac.uk>, Institute for Social and Economic Research, University of Essex, Colchester CO4 3SQ, U.K.
Philippe Van Kerm <philippe.vankerm@ceps.lu>, CEPS/INSTEAD, Differdange, Luxembourg.
Reference
Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken, NJ: John Wiley.
Also see
Online: help for smfit, dagumfit, gb2fit, lognfit, hillp if installed.