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# Discrete Weibull distribution

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

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 cmf ${\displaystyle 1-q^{(x+1)^{\beta }}}$ 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]