Power Generalised Weibull Distribution

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Published

January 14, 2024

1 The Power Generalised Weibull Distribution

The Power Generalised Weibull (PGW) distribution (Nikulin and Haghighi 2009) is a three-parameter distribution with support on \({\mathbb R}_+\). The corresponding hazard function can accommodate bathtub, unimodal and monotone (increasing and decreasing) hazard shapes. The PGW distribution has become popular in survival analysis given the tractability of its hazard and survival functions. Other flexible distributions that can account for these hazard shapes are discussed in Rubio et al. (2021) and Jones and Noufaily (2015).

1.1 Probability Density Function

The pdf of the PGW distribution is

\[f(t;\sigma,\nu,\gamma) = \dfrac{\nu}{\gamma \sigma^\nu}t^{\nu-1} \left[ 1 + \left(\dfrac{t}{\sigma}\right)^\nu\right]^{\left(\frac{1}{\gamma}-1\right)} \exp\left\{ 1- \left[ 1 + \left(\dfrac{t}{\sigma}\right)^\nu\right]^{\frac{1}{\gamma}} \right\},\]

where \(\sigma>0\) is a scale parameter, and \(\nu,\gamma >0\) are shape parameters.

1.2 Survival Function

The survival function of the PGW distribution is

\[S(t;\sigma,\nu,\gamma) = \exp\left\{ 1- \left[ 1 + \left(\dfrac{t}{\sigma}\right)^\nu\right]^{\frac{1}{\gamma}} \right\}.\]

1.3 Hazard Function

The hazard function of the PGW distribution is

\[h(t;\sigma,\nu,\gamma) = \dfrac{\nu}{\gamma \sigma^\nu}t^{\nu-1} \left[ 1 + \left(\dfrac{t}{\sigma}\right)^\nu\right]^{\left(\frac{1}{\gamma}-1\right)}.\] The cdf can be obtained as \(F(t;\sigma,\nu,\gamma)=1-S(t;\sigma,\nu,\gamma)\), and the cumulative hazard function as \(H(t;\sigma,\nu,\gamma) = -\log S(t;\sigma,\nu,\gamma)\), as usual.

1.4 Quantile Function

The quantile function of the PGW distribution is

\[Q(p;\sigma,\nu,\gamma) = \sigma \left[ \left( 1 - \log(1-p) \right)^{\gamma} - 1 \right]^{\frac{1}{\nu}},\]

where \(p\in(0,1)\).

The following Julia code shows the implementation of the pdf, survival function, hazard function, cumulative hazard function, quantile function, and random number generation associated to the PGW distribution using the Julia package HazReg.jl. Some illustrative examples are also presented.

2 Required packages

Code

using Distributions
using Random
using Plots
using StatsBase
using HazReg

3 Examples

3.1 Random number generation

Code

#= Fix the seed =#
Random.seed!(123)
#= True values of the parameters =#
sigma0 = 1
nu0 = 3
gamma0 = 2
#= Simulation =#
sim = randPGW(1000, sigma0, nu0, gamma0);

3.2 Some plots

Code

#= Histogram and probability density function =#
histogram(sim, normalize=:pdf, color=:gray, 
          bins = range(0, 5, length=30), label = "")
plot!(t -> pdfPGW(t, sigma0, nu0, gamma0),
      xlabel = "x", ylabel = "Density", title = "PGW distribution",
    xlims = (0,5),   xticks = 0:1:5, label = "", 
    xtickfont = font(16, "Courier"),  ytickfont = font(16, "Courier"),
    xguidefontsize=18, yguidefontsize=18, linewidth=3,
    linecolor = "blue")

Code

#= Empirical CDF and CDF =#

#= Empirical CDF=#
ecdfsim = ecdf(sim)

#= ad hoc CDF =#
function cdfPGW(t, sigma, nu, gamma) 
        val = 1 .- ccdfPGW.(t, sigma, nu, gamma)
        return val
end

plot(x -> ecdfsim(x), 0, 5, label = "ECDF", linecolor = "gray", linewidth=3)
plot!(t -> cdfPGW(t, sigma0, nu0, gamma0),
      xlabel = "x", ylabel = "CDF vs. ECDF", title = "PGW distribution",
    xlims = (0,5),   xticks = 0:1:5, label = "CDF", 
    xtickfont = font(16, "Courier"),  ytickfont = font(16, "Courier"),
    xguidefontsize=18, yguidefontsize=18, linewidth=3,
    linecolor = "blue")

Code

#= Hazard function =#
plot(t -> hPGW(t, 0.5, 2, 5),
      xlabel = "x", ylabel = "Hazard", title = "PGW distribution",
    xlims = (0,10),   xticks = 0:1:10, label = "", 
    xtickfont = font(16, "Courier"),  ytickfont = font(16, "Courier"),
    xguidefontsize=18, yguidefontsize=18, linewidth=3,
    linecolor = "blue")

References

Jones, M. C., and A. Noufaily. 2015. “Log-Location-Scale-Log-Concave Distributions for Survival and Reliability Analysis.” Electronic Journal of Statistics 9 (2): 2732–50.

Nikulin, M., and F. Haghighi. 2009. “On the Power Generalized Weibull Family: Model for Cancer Censored Data.” Metron – International Journal of Statistics 67 (1): 75–86.

Rubio, F. J., B. Rachet, B. Giorgi, C. Maringe, and A. Belot. 2021. “On Models for the Estimation of the Excess Mortality Hazard in Case of Insufficiently Stratified Life Tables.” Biostatistics 22 (1): 51–67.