Discrete Weibull distribution
None
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
making the cumulative distribution function
with
. Setting
makes the relationship with the geometric distribution apparent.
[1]
An alternative parametrization — related to the
Pareto distribution
— has been used to estimate parameters in
infectious disease modelling
.
[2]
This parametrization introduces a parameter
, meaning that the term
can be replaced with
. Therefore, the probability mass function can be expressed as
-
,
and the cumulative mass function can be expressed as
-
.
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 cumulative mass function:
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. [3]
See also
References
- ↑ Nakagawa, Toshio; Osaki, Shunji (1975). "The discrete Weibull distribution". IEEE Transactions on Reliability . 24 (5): 300–301. doi : 10.1109/TR.1975.5214915 . S2CID 6149392 .
- ↑ Endo A, Murayama H, Abbott S, et al. (2022). "Heavy-tailed sexual contact networks and monkeypox epidemiology in the global outbreak, 2022" . Science . 378 (6615): 90–94. doi : 10.1126/science.add4507 . PMID 36137054 .
- ↑ 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 .
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