a numeric vector of observations, or an object resulting from a call to an
estimating function that assumes a Pareto distribution
(e.g., epareto). If x is a numeric vector,
missing (NA), undefined (NaN), and infinite (Inf, -Inf)
values are allowed but will be removed.
p
numeric vector of probabilities for which quantiles will be estimated.
All values of p must be between 0 and 1. The default value is p=0.5.
method
character string specifying the method of estimating the distribution parameters.
Possible values are
"mle" (maximum likelihood; the default), and "lse" (least-squares).
See the DETAILS section of the help file for epareto for more
information on these estimation methods.
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
plot.pos.con=0.375. This argument is used only when method="lse".
digits
an integer indicating the number of decimal places to round to when printing out
the value of 100*p. The default value is digits=0.
Details
The function eqpareto returns estimated quantiles as well as
estimates of the location and scale parameters.
Quantiles are estimated by 1) estimating the location and scale parameters by
calling epareto, and then 2) calling the function
qpareto and using the estimated values for
location and scale.
Value
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.
Note
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.
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
epareto, Pareto, estimate.object.
Examples
# 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)
Results
R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)
R is free software and comes with ABSOLUTELY NO WARRANTY.
You are welcome to redistribute it under certain conditions.
Type 'license()' or 'licence()' for distribution details.
R is a collaborative project with many contributors.
Type 'contributors()' for more information and
'citation()' on how to cite R or R packages in publications.
Type 'demo()' for some demos, 'help()' for on-line help, or
'help.start()' for an HTML browser interface to help.
Type 'q()' to quit R.
> library(EnvStats)
Attaching package: 'EnvStats'
The following objects are masked from 'package:stats':
predict, predict.lm
The following object is masked from 'package:base':
print.default
> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/EnvStats/eqpareto.Rd_%03d_medium.png", width=480, height=480)
> ### Name: eqpareto
> ### Title: Estimate Quantiles of a Pareto Distribution
> ### Aliases: eqpareto
> ### Keywords: distribution htest
>
> ### ** Examples
>
> # 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
>
> #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)
>
>
>
>
>
> dev.off()
null device
1
>