ParetoUC: The Pareto Distribution

In VGAM: Vector Generalized Linear and Additive Models

The Pareto Distribution

Description

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.

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.

Value

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

Author(s)

T. W. Yee and Kai Huang

References

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

See Also

paretoff, ParetoIV.

Examples

alpha <- 3; k <- exp(1); x <- seq(2.8, 8, len = 300)
## Not run: 
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")

## End(Not run)
pvec <- seq(0.1, 0.9, by = 0.1)
qvec <- qpareto(pvec, scale = alpha, shape = k)
ppareto(qvec, scale = alpha, shape = k)
qpareto(ppareto(qvec, scale = alpha, shape = k),
        scale = alpha, shape = k) - qvec  # Should be 0

VGAM documentation built on Sept. 18, 2024, 9:09 a.m.