EnvStats (version 2.3.1)

epareto: 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 "mle" (maximum likelihood; the default), and "lse" (least-squares). See the DETAILS section 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".

Value

a list of class "estimate" containing the estimated parameters and other information. See estimate.object for details.

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).

Least-Squares Estimation (method="lse") The least-squares 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 least-squares 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).

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

Pareto.

Examples

# NOT RUN {
  # 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)
# }