# beniniUC: The Benini Distribution In VGAM: Vector Generalized Linear and Additive Models

 Benini R Documentation

## The Benini Distribution

### Description

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

### Usage

``````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`.

### 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.

### Value

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

### Author(s)

T. W. Yee and Kai Huang

### References

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

`benini1`.

### Examples

``````## Not run:
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

## End(Not run)
``````

VGAM documentation built on Sept. 19, 2023, 9:06 a.m.