Proof of Conjectures on the Standard Deviation, Skewness and Kurtosis of the Shifted Gompertz Distribution

Abstract

Three conjectures on the standard deviation, skewness and kurtosis of the shifted Gompertz distribution, as the shape parameter increases to +∞, are proved. In this regard, the exponential integral function and polygamma functions are used in the proofs. In addition, an explicit expression for the ith moment of this probabilistic model is obtained. These results allow to place the shifted Gompertz distribution in the Skewness–Kurtosis diagram, providing a valuable help in the decision to choose the shifted Gompertz distribution among the models to fit data. Their usefulness is illustrated by fitting a real malaria data set using the maximum likelihood method for estimating the parameters of the shifted Gompertz distribution and some classical models. Goodness-of-fit measures are used to compare their performance.