# Discrete Weibull distribution

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.

Parameters |
scale shape | ||
---|---|---|---|

Support | |||

pmf | |||

CDF |

## Alternative parametrizations

In the original paper by Nakagawa and Osaki they used the parametrization making the cmf with . Setting 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 where (unconventionally) and define parameters and . By replacing in the cmf:

We see that we get a location-scale parametrization:

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]

## References

- Nakagawa, Toshio; Osaki, Shunji (1975). "The discrete Weibull distribution".
*IEEE Transactions on Reliability*.**24**: 300–301. - Scholz, Fritz (1996). "Maximum Likelihood Estimation for Type I Censored Weibull Data Including Covariates".
*ISSTECH-96-022, Boeing Information & Support Services*. Retrieved 26 April 2016.