# Discrete Weibull distribution

Parameters $\alpha >0$ scale $\beta >0$ shape $x\in \{0,1,2,\ldots \}$ $\exp \left[-\left({\frac {x}{\alpha }}\right)^{\beta }\right]-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]$ $1-\exp \left[-\left({\frac {x+1}{\alpha }}\right)^{\beta }\right]$ 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 $q=e^{-\alpha ^{-\beta }}$ making the cmf $1-q^{(x+1)^{\beta }}$ with $q\in (0,1)$ . Setting $\beta =1$ makes the relationship with the geometric distribution apparent.

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

$\Pr(X\leq x)=\Pr(X\leq e^{y}-1).$ We see that we get a location-scale parametrization:

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