RTMBdist (version 1.0.1)

pgweibull: Power generalized Weibull distribution

Description

Survival, hazard, cumulative distribution, density, quantile and sampling function for the power generalized Weibull (PgW) distribution with parameters scale, shape and powershape.

Usage

spgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)
hpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)
ppgweibull(x, scale = 1, shape = 1, powershape = 1,
           lower.tail = TRUE, log.p = FALSE)
dpgweibull(x, scale = 1, shape = 1, powershape = 1, log = FALSE)
qpgweibull(p, scale = 1, shape = 1, powershape = 1)
rpgweibull(n, scale = 1, shape = 1, powershape = 1)

Value

dpgweibull gives the density, ppgweibull gives the distribution function, qpgweibull gives the quantile function, and rpgweibull generates random deviates. spgweibull gives the survival function and hpgweibull gives the hazard function.

Arguments

x

vector of quantiles

scale

positive scale parameter

shape

positive shape parameter

powershape

positive power shape parameter

log, log.p

logical; if TRUE, probabilities/ densities \(p\) are returned as \(\log(p)\).

lower.tail

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

p

vector of probabilities

n

number of observations

Details

The survival function of the PgW distribution is: $$ S(x) = \exp \left\{ 1 - \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma}} \right\}. $$ The hazard function is $$ \frac{\nu}{\gamma\theta^{\nu}}\cdot x^{\nu-1}\cdot \left[ 1 + \left(\frac{x}{\theta}\right)^{\nu}\right]^{\frac{1}{\gamma-1}} $$ The cumulative distribution function is then \(F(x) = 1 - S(x)\) and the density function is \(S(x)\cdot h(x)\).

If both shape parameters equal 1, the PgW distribution reduces to the exponential distribution (see dexp) with \(\texttt{rate} = 1/\texttt{scale}\) If the power shape parameter equals 1, the PgW distribution simplifies to the Weibull distribution (see dweibull) with the same parametrization.

Examples

x <- rpgweibull(1, 2, 2, 3)
d <- dpgweibull(x, 2, 2, 3)
p <- ppgweibull(x, 2, 2, 3)
q <- qpgweibull(p, 2, 2, 3)
s <- spgweibull(x, 2, 2, 3)
h <- hpgweibull(x, 2, 2, 3)