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. Alternative parametrizations. In the original paper by Nakagawa and Osaki they used the parametrization formula_1 making the cdf formula_2 with formula_3. Setting formula_4 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 formula_5 where (unconventionally ) formula_6 and define parameters formula_7 and formula_8. By replacing formula_9 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.