The Exponentiated Danish Distribution

Density, distribution function, quantile function, survival function, hazard function and random generation for the three-parameter Exponentiated Danish distribution, the \(a = 1\) submodel of the Beta-Danish family.

Usage

ded(x, b, c, k, log = FALSE)

ped(q, b, c, k, lower.tail = TRUE, log.p = FALSE)

qed(p, b, c, k, lower.tail = TRUE, log.p = FALSE)

red(n, b, c, k)

sed(x, b, c, k, log = FALSE)

hed(x, b, c, k, log = FALSE)

Arguments

x, q

Vector of quantiles.

b, c, k

Shape, shape and scale parameters.

log, log.p

Logical; return values on the log scale.

lower.tail

Logical; if `TRUE` (default) probabilities are \(P[X \le x]\).

p

Vector of probabilities.

n

Number of observations to generate.

Value

Vectors of the same form as the Beta-Danish equivalents.

Details

These are thin wrappers that fix \(a = 1\), provided because the ED submodel is a named distribution in its own right in the underlying work and writing `dbetadanish(x, 1, b, c, k)` obscures that.

At \(a = 1\) the beta generator collapses and the distribution function simplifies to \(F(t) = 1 - \{1 - G(t)\}^{b}\) with \(G(t) = \{kt/(1+kt)\}^{c}\). The upper tail still has index \(-b\), so the moment condition \(b > r\) is unchanged.

See also

[BetaDanish] for the four-parameter parent, [fit_betadanish()] with `submodel = TRUE` for estimation.

Examples

ded(2, b = 3, c = 2, k = 1)
#> [1] 0.1371742
ped(2, b = 3, c = 2, k = 1)
#> [1] 0.8285322
qed(0.5, b = 3, c = 2, k = 1)
#> [1] 0.8321797
red(5, b = 3, c = 2, k = 1)
#> [1] 2.4098339 0.3318704 0.1203918 0.5327162 0.6585504

# Identical to the a = 1 parent
all.equal(ded(2, 3, 2, 1), dbetadanish(2, 1, 3, 2, 1))
#> [1] TRUE