Distributions Derived from the Continuous Iteration of the Hyperbolic Sine Function

Abstract—Families of distributions built from the fractional or continuous iteration of exponential-type functions are characterized by a wide range of tail-heaviness. The present paper aims to define classes of distributions supported on the whole real line based on the continuous iteration of the hyperbolic sine function sinh. This function has already been commonly employed in univariate transformations such as the Johnson’s $S_{U}$ and sinh–arcsinh transforms. The tail versatility generated by a transformation based on the continuous iteration of sinh is highlighted based on an initial logistic distribution. It leads to the Hyperbolic Tetration distribution. The Double Hyperbolic Tetration distribution, defined from two successive hyperbolic transformations, is also introduced. It is among the first class of distributions with potential distinct tetration indices at plus and minus infinity. The distributions are applied to multiple data sets in hydrology.