Discrete Weibull

Parameters	${\displaystyle \alpha >0}$ scale ${\displaystyle \beta >0}$ shape
Support	${\displaystyle x\in \{0,1,2,\ldots \))$
PMF	${\displaystyle \exp \left[-\left({\frac {x}{\alpha ))\right)^{\beta }\right]-\exp \left[-\left({\frac {x+1}{\alpha ))\right)^{\beta }\right]}$
CDF	${\displaystyle 1-\exp \left[-\left({\frac {x+1}{\alpha ))\right)^{\beta }\right]}$

In probability theory and statistics, the discrete Weibull distribution is the discrete variant of the Weibull distribution. It was first described by Nakagawa and Osaki in 1975.

Alternative parametrizations

In the original paper by Nakagawa and Osaki they used the parametrization ${\displaystyle q=e^{-\alpha ^{-\beta ))}$ making the cdf ${\displaystyle 1-q^{(x+1)^{\beta ))}$ with ${\displaystyle q\in (0,1)}$. Setting ${\displaystyle \beta =1}$ makes the relationship with the geometric distribution apparent.^[1]

Location-scale transformation

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 cmf:

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

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. ^[2]