Piecewise constant hazard distributions

Definition

The PiecewiseConstantHazardDistribution is one of the most simple and yet most usefull distribution provided in this package. These distributions are defined by their hazard functions, which are assumed to be piecewise constant (hence their names).

While dealing with census data and rate tables, having a survival model defined by a piecewise constant hazard is very common. In particular, random lifes extracted from RateTables from RateTables.jl follows this pattern.

Examples

using SurvivalDistributions, Distributions, Random, Plots, StatsBase
Random.seed!(123)
∂t = rand(20)
λ = rand(20)
D = PiecewiseConstantHazardDistribution(∂t,λ)
sim = rand(D,1000);

1000-element Vector{Float64}:
 3.3156618965693783
 1.940555419606624
 1.9334156799858147
 1.0932388990150501
 0.618833597638565
 0.050478867220641876
 1.0877081196502922
 1.3367500039352507
 7.8606743940199495
 0.17606520023347771
 ⋮
 1.6029796590109677
 1.8448864039253583
 1.5818278138226063
 1.3340378604150012
 4.78353551176096
 0.36163673342660474
 0.536170464409816
 0.42414369434832033
 0.4210414888055914

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

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

As excepted, it is quite random.

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))

The comparison is not too bad ! 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))