Estimate Quantiles of a Pareto Distribution
Estimate quantiles of a Pareto distribution.
eqpareto(x, p = 0.5, method = "mle", plot.pos.con = 0.375, digits = 0)
x |
a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a Pareto distribution
(e.g., |
p |
numeric vector of probabilities for which quantiles will be estimated.
All values of |
method |
character string specifying the method of estimating the distribution parameters.
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
|
digits |
an integer indicating the number of decimal places to round to when printing out
the value of |
The function eqpareto returns estimated quantiles as well as
estimates of the location and scale parameters.
If x is a numeric vector, eqpareto returns a
list of class "estimate" containing the estimated quantile(s) and other
information. See estimate.object for details.
If x is the result of calling an estimation function, eqpareto
returns a list whose class is the same as x. The list
contains the same components as x, as well as components called
quantiles and quantile.method.
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 # and the 90'th percentile. # (Note: the call to set.seed simply allows you to reproduce this example.) set.seed(250) dat <- rpareto(30, location = 1, shape = 1) eqpareto(dat, p = 0.9) #Results of Distribution Parameter Estimation #-------------------------------------------- # #Assumed Distribution: Pareto # #Estimated Parameter(s): location = 1.009046 # shape = 1.079850 # #Estimation Method: mle # #Estimated Quantile(s): 90'th %ile = 8.510708 # #Quantile Estimation Method: Quantile(s) Based on # mle Estimators # #Data: dat # #Sample Size: 30 #---------- # Clean up #--------- rm(dat)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.