Sinh-arcsinh distributions: a broad family giving rise to powerful tests of normality and symmetry

We introduce the ‘sinh-arcsinh transformation’ and thence, by applying it to random variables from some ‘generating’ distribution with no further parameters beyond location and scale (which we take for most of the paper to be the normal), a new family of ‘sinh-arcsinh distributions’. This four parameter family has both symmetric and skewed members and allows for tailweights that are both heavier and lighter than those of the generating distribution. The ‘central’ place of the normal distribution in this family affords likelihood ratio tests of normality that appear to be superior to the state-of-the-art because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are of interest too. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson SU distributions while light-tailed symmetric sinh-arcsinh distributions behave like Rieck and Nedelman’s sinh-normal distributions, the sinh-arcsinh family allowing a seamless transition between the two, via the normal, controlled by a single parameter. The sinh-arcsinh family is very tractable and many properties are explored. Likelihood inference is pursued, including an attractive reparametrisation. A multivariate version is considered. Options and extensions are discussed.