The generalized Beta of the first kind Distribution

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Description

Density, distribution function, quantile function and random generation for the GB1 distribution with parameters shape0, shape1 and shape2.

Usage

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dgbeta(x, shape0, shape1, shape2, log = FALSE)
pgbeta(q, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE)
qgbeta(p, shape0, shape1, shape2, lower.tail = TRUE, log.p = FALSE)
rgbeta(n, shape0, shape1, shape2)
ecgbeta(x, shape0, shape1, shape2)
mgbeta(order, shape0, shape1, shape2)

Arguments

x, q

vector of quantiles.

p

vector of probabilities.

n

number of observations. If length(n) > 1, the length is taken to be the number required.

shape0, shape1, shape2

positive parameters of the GB1 distribution.

log, log.p

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x].

order

order of the raw moment.

Details

The GB1 distribution with parameters shape0 = g, shape1 = a and shape2 = b has density

Γ(a+b)/(Γ(a)Γ(b))x^(a/g-1)(1-x^{1/g})^(b-1)/g

for a,b,g > 0 and 0 ≤ x ≤ 1 where the boundary values at x=0 or x=1 are defined as by continuity (as limits).

Value

dgbeta gives the density, pgbeta the distribution function, qgbeta the quantile function, and rgbeta generates random deviates.

References

Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Abramowitz, M. and Stegun, I. A. (1972) Handbook of Mathematical Functions. New York: Dover. Chapter 6: Gamma and Related Functions.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, volume 2, especially chapter 25. Wiley, New York.

See Also

Distributions for other standard distributions.

Examples

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