epareto {EnvStats}  R Documentation 
Estimate Parameters of a Pareto Distribution
Description
Estimate the location and shape parameters of a Pareto distribution.
Usage
epareto(x, method = "mle", plot.pos.con = 0.375)
Arguments
x 
numeric vector of observations. 
method 
character string specifying the method of estimation. Possible values are

plot.pos.con 
numeric scalar between 0 and 1 containing the value of the plotting position
constant used to construct the values of the empirical cdf. The default value is

Details
If x
contains any missing (NA
), undefined (NaN
) or
infinite (Inf
, Inf
) values, they will be removed prior to
performing the estimation.
Let \underline{x} = (x_1, x_2, \ldots, x_n)
be a vector of
n
observations from a Pareto distribution with
parameters location=
\eta
and shape=
\theta
.
Maximum Likelihood Estimatation (method="mle"
)
The maximum likelihood estimators (mle's) of \eta
and \theta
are
given by (Evans et al., 1993; p.122; Johnson et al., 1994, p.581):
\hat{\eta}_{mle} = x_{(1)} \;\;\;\; (1)
\hat{\theta}_{mle} = n [\sum_{i=1}^n log(\frac{x_i}{\hat{\eta}_{mle}}) ]^{1} \;\;\;\; (2)
where x_(1)
denotes the first order statistic (i.e., the minimum value).
LeastSquares Estimation (method="lse"
)
The leastsquares estimators (lse's) of \eta
and \theta
are derived as
follows. Let X
denote a Pareto random variable with parameters
location=
\eta
and shape=
\theta
. It can be shown that
log[1  F(x)] = \theta log(\eta)  \theta log(x) \;\;\;\; (3)
where F
denotes the cumulative distribution function of X
. Set
y_i = log[1  \hat{F}(x_i)] \;\;\;\; (4)
z_i = log(x_i) \;\;\;\; (5)
where \hat{F}(x)
denotes the empirical cumulative distribution function
evaluated at x
. The leastsquares estimates of \eta
and \theta
are obtained by solving the regression equation
y_i = \beta_{0} + \beta_{1} z_i \;\;\;\; (6)
and setting
\hat{\theta}_{lse} = \hat{\beta}_{1} \;\;\;\; (7)
\hat{\eta}_{lse} = exp(\frac{\hat{\beta}_0}{\hat{\theta}_{lse}}) \;\;\;\; (8)
(Johnson et al., 1994, p.580).
Value
a list of class "estimate"
containing the estimated parameters and other information.
See estimate.object
for details.
Note
The Pareto distribution is named after Vilfredo Pareto (18481923), a professor
of economics. It is derived from Pareto's law, which states that the number of
persons N
having income \ge x
is given by:
N = A x^{\theta}
where \theta
denotes Pareto's constant and is the shape parameter for the
probability distribution.
The Pareto distribution takes values on the positive real line. All values must be
larger than the “location” parameter \eta
, which is really a threshold
parameter. There are three kinds of Pareto distributions. The one described here
is the Pareto distribution of the first kind. Stable Pareto distributions have
0 < \theta < 2
. Note that the r
'th moment only exists if
r < \theta
.
The Pareto distribution is related to the
exponential distribution and
logistic distribution as follows.
Let X
denote a Pareto random variable with location=
\eta
and
shape=
\theta
. Then log(X/\eta)
has an exponential distribution
with parameter rate=
\theta
, and log\{ [(X/\eta)^\theta]  1 \}
has a logistic distribution with parameters location=
0
and
scale=
1
.
The Pareto distribution has a very long righthand tail. It is often applied in the study of socioeconomic data, including the distribution of income, firm size, population, and stock price fluctuations.
Author(s)
Steven P. Millard (EnvStats@ProbStatInfo.com)
References
Forbes, C., M. Evans, N. Hastings, and B. Peacock. (2011). Statistical Distributions. Fourth Edition. John Wiley and Sons, Hoboken, NJ.
Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.
See Also
Examples
# Generate 30 observations from a Pareto distribution with parameters
# location=1 and shape=1 then estimate the parameters.
# (Note: the call to set.seed simply allows you to reproduce this example.)
set.seed(250)
dat < rpareto(30, location = 1, shape = 1)
epareto(dat)
#Results of Distribution Parameter Estimation
#
#
#Assumed Distribution: Pareto
#
#Estimated Parameter(s): location = 1.009046
# shape = 1.079850
#
#Estimation Method: mle
#
#Data: dat
#
#Sample Size: 30
#
# Compare the results of using the leastsquares estimators:
epareto(dat, method="lse")$parameters
#location shape
#1.085924 1.144180
#
# Clean up
#
rm(dat)