The Benini Distribution

Density, distribution function, quantile function and random generation for the Benini distribution with parameter shape.

dbenini(x, y0, shape, log = FALSE)
pbenini(q, y0, shape, lower.tail = TRUE, log.p = FALSE)
qbenini(p, y0, shape, lower.tail = TRUE, log.p = FALSE)
rbenini(n, y0, shape)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. Same as runif.

y0

the scale parameter \(y_0\).

shape

the positive shape parameter \(b\).

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

dbenini gives the density, pbenini gives the distribution function, qbenini gives the quantile function, and rbenini generates random deviates.

References

Kleiber, C. and Kotz, S. (2003). Statistical Size Distributions in Economics and Actuarial Sciences, Hoboken, NJ, USA: Wiley-Interscience.

Author

T. W. Yee and Kai Huang

Details

See benini1, the VGAM family function for estimating the parameter \(s\) by maximum likelihood estimation, for the formula of the probability density function and other details.

See also

benini1.

Examples

if (FALSE) { # \dontrun{
y0 <- 1; shape <- exp(1)
xx <- seq(0.0, 4, len = 101)
plot(xx, dbenini(xx, y0 = y0, shape = shape), col = "blue",
     main = "Blue is density, orange is the CDF", type = "l",
     sub = "Purple lines are the 10,20,...,90 percentiles",
     ylim = 0:1, las = 1, ylab = "", xlab = "x")
abline(h = 0, col = "blue", lty = 2)
lines(xx, pbenini(xx, y0 = y0, shape = shape), col = "orange")
probs <- seq(0.1, 0.9, by = 0.1)
Q <- qbenini(probs, y0 = y0, shape = shape)
lines(Q, dbenini(Q, y0 = y0, shape = shape),
      col = "purple", lty = 3, type = "h")
pbenini(Q, y0 = y0, shape = shape) - probs  # Should be all zero
} # }