Power generalized Weibull distribution
pgweibull.RdSurvival, 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)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
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.
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.