The Pareto(IV/III/II) Distributions

Density, distribution function, quantile function and random generation for the Pareto(IV/III/II) distributions.

dparetoIV(x, location = 0, scale = 1, inequality = 1, shape = 1,
          log = FALSE)
pparetoIV(q, location = 0, scale = 1, inequality = 1, shape = 1,
          lower.tail = TRUE, log.p = FALSE)
qparetoIV(p, location = 0, scale = 1, inequality = 1, shape = 1,
          lower.tail = TRUE, log.p = FALSE)
rparetoIV(n, location = 0, scale = 1, inequality = 1, shape = 1)
dparetoIII(x, location = 0, scale = 1, inequality = 1, log = FALSE)
pparetoIII(q, location = 0, scale = 1, inequality = 1,
           lower.tail = TRUE, log.p = FALSE)
qparetoIII(p, location = 0, scale = 1, inequality = 1,
           lower.tail = TRUE, log.p = FALSE)
rparetoIII(n, location = 0, scale = 1, inequality = 1)
dparetoII(x, location = 0, scale = 1, shape = 1, log = FALSE)
pparetoII(q, location = 0, scale = 1, shape = 1,
          lower.tail = TRUE, log.p = FALSE)
qparetoII(p, location = 0, scale = 1, shape = 1,
          lower.tail = TRUE, log.p = FALSE)
rparetoII(n, location = 0, scale = 1, shape = 1)
dparetoI(x, scale = 1, shape = 1, log = FALSE)
pparetoI(q, scale = 1, shape = 1,
         lower.tail = TRUE, log.p = FALSE)
qparetoI(p, scale = 1, shape = 1,
         lower.tail = TRUE, log.p = FALSE)
rparetoI(n, scale = 1, shape = 1)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as in runif.

location

the location parameter.

scale, shape, inequality

the (positive) scale, inequality and shape 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

Functions beginning with the letters d give the density, p give the distribution function, q give the quantile function, and r generates random deviates.

References

Brazauskas, V. (2003). Information matrix for Pareto(IV), Burr, and related distributions. Comm. Statist. Theory and Methods 32, 315–325.

Arnold, B. C. (1983). Pareto Distributions. Fairland, Maryland: International Cooperative Publishing House.

Author

T. W. Yee and Kai Huang

Details

For the formulas and other details see paretoIV.

Note

The functions [dpqr]paretoI are the same as [dpqr]pareto except for a slight change in notation: \(s=k\) and \(b=\alpha\); see Pareto.

See also

paretoIV, Pareto.

Examples

if (FALSE) { # \dontrun{
x <- seq(-0.2, 4, by = 0.01)
loc <- 0; Scale <- 1; ineq <- 1; shape <- 1.0
plot(x, dparetoIV(x, loc, Scale, ineq, shape), type = "l",
     main = "Blue is density, orange is the CDF", col = "blue",
     sub = "Purple are 5,10,...,95 percentiles", ylim = 0:1,
     las = 1, ylab = "")
abline(h = 0, col = "blue", lty = 2)
Q <- qparetoIV(seq(0.05, 0.95,by = 0.05), loc, Scale, ineq, shape)
lines(Q, dparetoIV(Q, loc, Scale, ineq, shape), col = "purple",
      lty = 3, type = "h")
lines(x, pparetoIV(x, loc, Scale, ineq, shape), col = "orange")
abline(h = 0, lty = 2)
} # }