The Pareto Distribution

Density, distribution function, quantile function and random generation for the Pareto(I) distribution with parameters scale and shape.

Usage

dpareto(x, scale = 1, shape, log = FALSE)
ppareto(q, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
qpareto(p, scale = 1, shape, lower.tail = TRUE, log.p = FALSE)
rpareto(n, scale = 1, shape)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as in runif.

scale, shape

the \(\alpha\) and \(k\) parameters.

log

Logical. If log = TRUE then the logarithm of the density is returned.

lower.tail, log.p

Same meaning as in pnorm or qnorm.

Value

dpareto gives the density, ppareto gives the distribution function, qpareto gives the quantile function, and rpareto generates random deviates.

References

Forbes, C., Evans, M., Hastings, N. and Peacock, B. (2011). Statistical Distributions, Hoboken, NJ, USA: John Wiley and Sons, Fourth edition.

Author

T. W. Yee and Kai Huang

Details

See paretoff, the VGAM family function for estimating the parameter \(k\) by maximum likelihood estimation, for the formula of the probability density function and the range restrictions imposed on the parameters.

Examples

alpha <- 3; k <- exp(1); x <- seq(2.8, 8, len = 300)
if (FALSE) { # \dontrun{
plot(x, dpareto(x, scale = alpha, shape = k), type = "l",
     main = "Pareto density split into 10 equal areas")
abline(h = 0, col = "blue", lty = 2)
qvec <- qpareto(seq(0.1, 0.9, by = 0.1), scale = alpha, shape = k)
lines(qvec, dpareto(qvec, scale = alpha, shape = k),
      col = "purple", lty = 3, type = "h")
} # }
pvec <- seq(0.1, 0.9, by = 0.1)
qvec <- qpareto(pvec, scale = alpha, shape = k)
ppareto(qvec, scale = alpha, shape = k)
#> [1] 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
qpareto(ppareto(qvec, scale = alpha, shape = k),
        scale = alpha, shape = k) - qvec  # Should be 0
#> [1] 0 0 0 0 0 0 0 0 0