Estimate Parameters of a Pareto Distribution
Estimate the location and shape parameters of a Pareto distribution.
epareto(x, method = "mle", plot.pos.con = 0.375)
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
|
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, …, x_n) be a vector of
n observations from a Pareto distribution with
parameters location=η and shape=θ.
Maximum Likelihood Estimatation (method="mle")
The maximum likelihood estimators (mle's) of η and θ are
given by (Evans et al., 1993; p.122; Johnson et al., 1994, p.581):
\hat{η}_{mle} = x_{(1)} \;\;\;\; (1)
\hat{θ}_{mle} = n [∑_{i=1}^n log(\frac{x_i}{\hat{η}_{mle}}) ]^{-1} \;\;\;\; (2)
where x_(1) denotes the first order statistic (i.e., the minimum value).
Least-Squares Estimation (method="lse")
The least-squares estimators (lse's) of η and θ are derived as
follows. Let X denote a Pareto random variable with parameters
location=η and shape=θ. It can be shown that
log[1 - F(x)] = θ log(η) - θ 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 least-squares estimates of η and θ are obtained by solving the regression equation
y_i = β_{0} + β_{1} z_i \;\;\;\; (6)
and setting
\hat{θ}_{lse} = -\hat{β}_{1} \;\;\;\; (7)
\hat{η}_{lse} = exp(\frac{\hat{β}_0}{\hat{θ}_{lse}}) \;\;\;\; (8)
(Johnson et al., 1994, p.580).
a list of class "estimate" containing the estimated parameters and other information.
See estimate.object for details.
The Pareto distribution is named after Vilfredo Pareto (1848-1923), a professor of economics. It is derived from Pareto's law, which states that the number of persons N having income ≥ x is given by:
N = A x^{-θ}
where θ 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 η, 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 < θ < 2. Note that the r'th moment only exists if r < θ.
The Pareto distribution is related to the
exponential distribution and
logistic distribution as follows.
Let X denote a Pareto random variable with location=η and
shape=θ. Then log(X/η) has an exponential distribution
with parameter rate=θ, and -log\{ [(X/η)^θ] - 1 \}
has a logistic distribution with parameters location=0 and
scale=1.
The Pareto distribution has a very long right-hand tail. It is often applied in the study of socioeconomic data, including the distribution of income, firm size, population, and stock price fluctuations.
Steven P. Millard (EnvStats@ProbStatInfo.com)
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.
# 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 least-squares estimators: epareto(dat, method="lse")$parameters #location shape #1.085924 1.144180 #---------- # Clean up #--------- rm(dat)
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