The unit-Weibull distribution (UW) is a continuous probability distribution with domain on . Useful for indices and rates, or bounded variables with a domain. It was originally proposed by Mazucheli et al using a transformation of the Weibull distribution.
Its probability density function is defined as:
And its cumulative distribution function is:
The quantile function of the UW distribution is given by:
Having a closed form expression for the quantile function, may make it a more flexible alternative for a quantile regression model against the classical Beta regression model.
The th raw moment of the UW distribution can be obtained through:
The skewness and kurtosis measures can be obtained upon substituting the raw moments from the expressions:
The hazard rate function of the UW distribution is given by:
Let be a random sample of size from the UW distribution with probability density function defined before. Then, the log-likelihood function of is:
The likelihood estimate of is obtained by solving the non-linear equations
and
The expected Fisher information matrix of based on a single observation is given by
where and is the EulerâÂÂs constant.
When , follows the power function distribution and the th raw moment of the UW distribution becomes:
In this case, the mean, variance, skewness and kurtosis, are:
The skewness can be negative, zero, or positive when . And if , with , follows the standard uniform distribution, and the measures becomes:
For the case of , follows the unit-Rayleigh distribution, and:
where
Is the complementary error function. In this case, the measures of the distribution are:
It was shown to outperform, against other distributions, like the Beta and Kumaraswamy distributions, in: maximum flood level, petroleum reservoirs, risk management cost effectiveness, and recovery rate of CD34+cells data.