Discrete Weibull distribution

The original paper by Nakagawa and Osaki used the parametrization ${\displaystyle p=q^{k^{-\beta }}}$ making the cumulative distribution function ${\displaystyle 1-q^{(x+1)^{\beta }}}$with ${\displaystyle q\in (0,1)}$ and the probability mass function ${\displaystyle q^{k^{\beta }}-q^{(k+1)^{\beta }}}$. Setting ${\displaystyle \beta =1}$ makes the relationship with the geometric distribution apparent.^[2]

An alternative parametrization — related to the Pareto distribution — has been used to estimate parameters in infectious disease modelling.^[3] This parametrization introduces a parameter ${\displaystyle \kappa ={\frac {\beta }{\alpha ^{\beta }}}}$, meaning that the term ${\displaystyle \left({\frac {1}{\alpha }}\right)^{\beta }}$ can be replaced with ${\displaystyle {\frac {\kappa }{\beta }}}$. Therefore, the probability mass function can be expressed as

${\displaystyle \exp \left[-{\frac {\kappa x^{\beta }}{\beta }}\right]-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}$,

and the cumulative mass function can be expressed as

${\displaystyle 1-\exp \left[-{\frac {\kappa \left(x+1\right)^{\beta }}{\beta }}\right]}$.

The continuous Weibull distribution has a close relationship with the Gumbel distribution which is easy to see when log-transforming the variable. A similar transformation can be made on the discrete Weibull.

Define ${\displaystyle e^{Y}-1=X}$ where (unconventionally) ${\displaystyle Y=\log(X+1)\in \{\log(1),\log(2),\ldots \}}$ and define parameters ${\displaystyle \mu =\log(\alpha )}$ and ${\displaystyle \sigma ={\frac {1}{\beta }}}$. By replacing ${\displaystyle x}$ in the cumulative mass function:

${\displaystyle \Pr(X\leq x)=\Pr(X\leq e^{y}-1).}$

We see that we get a location-scale parametrization:

${\displaystyle =1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]=1-\exp \left[-\left({\frac {e^{y}}{e^{\mu }}}\right)^{\frac {1}{\sigma }}\right]=1-\exp \left[-\exp \left[{\frac {y-\mu }{\sigma }}\right]\right]}$

which in estimation settings makes a lot of sense. This opens up the possibility of regression with frameworks developed for Weibull regression and extreme-value-theory.^[4]

Examples of applications of the discrete Weibull distribution include:

• Modeling time to failure over discrete time units^[2]
• Modeling count data^[5]
• Modeling infectious disease^[3]

• Weibull distribution
• Geometric distribution
• q-Weibull distribution