Exponentiated Weibull Distribution

Definition

The Exponentiated Weibull distribution is obtain by exponentiating the cdf of the Weibull distribution. This simple transformation adds a second shape parameter that, interestingly, induces a lot of flexibility on the hazard function. The hazard function of the Exponentiated Weibull distribution can capture the basic shapes: constant, increasing, decreasing, bathtub, and unimodal, making it appealing for survival models.

The probability density function and cumulative distribution function of the Exponentiated Weibull distribution are respectively given by:

\[\begin{split} f(t) &= \alpha \dfrac{\kappa}{\lambda} \left(\dfrac{t}{\lambda}\right)^{\kappa-1} \left[1-\exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}\right]^{\alpha-1} \exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}, \\ F(t) &= \left[1-\exp\left\{-\left(\dfrac{t}{\lambda}\right)^{\kappa}\right\}\right]^{\alpha}, \end{split}\]

where $t>0$, $\alpha>0$, $\lambda>0$, and $\kappa>0$. The parameter $\lambda$ is a scale parameter, $\kappa$ is a shape parameter, and $\alpha$ is the power (shape) parameter.

Examples

Let us sample a dataset from an Exponentiated Weibull distribution:

using SurvivalDistributions, Distributions, Random, Plots, StatsBase
Random.seed!(123)
D = ExponentiatedWeibull(0.25, 0.5, 5)
sim = rand(D,1000);

1000-element Vector{Float64}:
    9.765271462065016
   13.7837598460054
  101.97641758285445
    1.2847442353531975
    9.99578064534659
    4.838506975877864
    0.17656401895523055
  169.41690560685615
   13.336290531528054
    3.345811394332756
    ⋮
    0.5899401809183444
 1131.2001975660924
  227.44425919326252
   57.609454946022126
    5.098260392504398
  134.72464474364648
 1250.872530553925
   24.62918690981065
    1.0273374308774073

First, let's have a look at the hazard function:

plot(t -> hazard(D,t), ylabel = "Hazard", xlims = (0,10))

Then, we can verify the coherence of our code by comparing the obtained sample and the true pdf:

histogram(sim, normalize=:pdf, bins = range(0, 5, length=30))
plot!(t -> pdf(D,t), ylabel = "Density", xlims = (0,5))

We could also compare the empirical and theroetical cdfs:

ecdfsim = ecdf(sim)
plot(x -> ecdfsim(x), 0, 5, label = "ECDF", linecolor = "gray", linewidth=3)
plot!(t -> cdf(D,t), xlabel = "x", ylabel = "CDF vs. ECDF", xlims = (0,5))