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)

Arguments

x: 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.

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

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 $\ge x$ 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 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.

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)