EnvStats (version 2.7.0)

Pareto: The Pareto Distribution

Description

Density, distribution function, quantile function, and random generation for the Pareto distribution with parameters location and shape.

Usage

dpareto(x, location, shape = 1)
  ppareto(q, location, shape = 1)
  qpareto(p, location, shape = 1)
  rpareto(n, location, shape = 1)

Value

dpareto gives the density, ppareto gives the distribution function,

qpareto gives the quantile function, and rpareto generates random deviates.

Arguments

x

vector of quantiles.

q

vector of quantiles.

p

vector of probabilities between 0 and 1.

n

sample size. If length(n) is larger than 1, then length(n) random values are returned.

location

vector of (positive) location parameters.

shape

vector of (positive) shape parameters. The default is shape=1.

Author

Steven P. Millard ()

Details

Let \(X\) be a Pareto random variable with parameters location=\(\eta\) and shape=\(\theta\). The density function of \(X\) is given by: $$f(x; \eta, \theta) = \frac{\theta \eta^\theta}{x^{\theta + 1}}, \; \eta > 0, \; \theta > 0, \; x \ge \eta$$ The cumulative distribution function of \(X\) is given by: $$F(x; \eta, \theta) = 1 - (\frac{\eta}{x})^\theta$$ and the \(p\)'th quantile of \(X\) is given by: $$x_p = \eta (1 - p)^{-1/\theta}, \; 0 \le p \le 1$$ The mode, mean, median, variance, and coefficient of variation of \(X\) are given by: $$Mode(X) = \eta$$ $$E(X) = \frac{\theta \eta}{\theta - 1}, \; \theta > 1$$ $$Median(X) = x_{0.5} = 2^{1/\theta} \eta$$ $$Var(X) = \frac{\theta \eta^2}{(\theta - 1)^2 (\theta - 1)}, \; \theta > 2$$ $$CV(X) = [\theta (\theta - 2)]^{-1/2}, \; \theta > 2$$

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

epareto, eqpareto, Exponential, Probability Distributions and Random Numbers.

Examples

  # Density of a Pareto distribution with parameters location=1 and shape=1, 
  # evaluated at 2, 3 and 4: 

  dpareto(2:4, 1, 1) 
  #[1] 0.2500000 0.1111111 0.0625000

  #----------

  # The cdf of a Pareto distribution with parameters location=2 and shape=1, 
  # evaluated at 3, 4, and 5: 

  ppareto(3:5, 2, 1) 
  #[1] 0.3333333 0.5000000 0.6000000

  #----------

  # The 25'th percentile of a Pareto distribution with parameters 
  # location=1 and shape=1: 

  qpareto(0.25, 1, 1) 
  #[1] 1.333333

  #----------

  # A random sample of 4 numbers from a Pareto distribution with parameters 
  # location=3 and shape=2. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(10) 
  rpareto(4, 3, 2)
  #[1] 4.274728 3.603148 3.962862 5.415322