Comparison of five estimation methods for the parameters of the Johnson unbounded distribution using simulated and real-data samples

As reported by several authors, for some samples from a Johnson unbounded (SU) distribution, the log-likelihood function does not have a local maximum with respect to the shift and scale parameters or may not satisfy the required regularity conditions to achieve the asymptotic efficiency of the maximum likelihood (ML) method for parameter estimation. This non-regularity of the likelihood function caused occasional non-convergence of algorithms to apply the ML method to estimate the parameters of a Johnson SU distribution. This is why there has been several alternative proposals to estimate these parameters, including the four-quantile matching rule of Slifker and Shapiro, a method based on moments proposed by Tuenter, and a method based on ML and regression proposed by George and Ramachandran. However, all the above-mentioned methods need some conditions on the sample to fit the Johnson SU distribution. In this article, we report the C++ implementation to fit a Johnson SU distribution, and the empirical comparison of the methods of ML, Slifker-Shapiro, Tuenter and George and Ramachandran, plus an implementation based on the minimization of the Cramér-von Mises distance. We present experimental results that show that the implementation based on minimum Cramér-von Mises distance performs very well, with apparently no requirements to produce reasonable estimates, achieving lower bias than the ML method for small sample sizes.