Summary. 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 S U 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.

We propose a new family of symmetric unimodal distributions on the circle which contains the uniform, von Mises, cardioid and wrapped Cauchy distributions, amongst others, as special cases. The basic form of the densities of this family is very simple, although its normalisation constant involves an associated Legendre function. The family of distributions can also be derived by conditioning and projecting certain bivariate spherically and elliptically symmetric distributions on to the circle. Trigonometric moments are available and circular variance and kurtosis are investigated in detail. The new family is compared with the wrapped normal and wrapped symmetric stable distributions. Theoretical aspects of maximum likelihood estimation are considered, and likelihood is used to fit the family of distributions to two example sets of data. The problem of the sample sizes necessary in practice to reliably discriminate between the von Mises, cardioid and wrapped Cauchy distributions is investigated. Finally, extension to a family of rotationally symmetric distributions on the sphere is briefly made.

This paper considers various unresolved inference problems for the skewnormal distribution. We give reasons as to why the direct parameterization should not be used as a general basis for estimation, and consider method of moments and maximum likelihood estimation for the distribution's centred parameterization. Large sample theory results are given for the method of moments estimators, and numerical approaches for obtaining maximum likelihood estimates are discussed. Simulation is used to assess the performance of the two types of estimation. We also present procedures for testing for departures from the limiting folded normal distribution. Data on the percentage body fat of elite athletes are used to illustrate some of the issues raised.