The unit-Gompertz distribution (UGo) is a continuous probability distribution with domain on . Useful for bounded variables with a domain. It was originally proposed by Mazucheli et al using a transformation of the Gompertz distribution. It is a suitable alternative for fitting skewed data. A quantile regression model has also been derived from the distribution.
Definitions
Probability density function Its probability density function is defined as:
Cumulative distribution function And its cumulative distribution function is:
Quantile function The quantile function of the UGo distribution is given by:
Properties Moments
The th raw moment of the UGo distribution can be obtained through:
where is the upper incomplete gamma function. The moments exists only when .
Skewness and kurtosis The skewness and kurtosis measures can be obtained upon substituting the raw moments from the expressions:
Hazard rate The hazard rate function of the UGo distribution is given by:
Parameter estimation Let be a random sample of size from the UGo 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 Hessian matrix of the log-likelihood with respect to the unknown parameters is given by
The expected Fisher information matrix of based on a single observation is given by
where represents the exponential integral function defined as
and
Applications
It was shown to outperform, against other distributions, like the beta and Kumaraswamy distributions, in: maximum flood level, literacy rates, and stress-strength reliability estimation data.
See also
Gompertz distribution
References