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Biometrika Advance Access originally published online on October 25, 2009
Biometrika 2009 96(4):761-780; doi:10.1093/biomet/asp053
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© 2009 Biometrika Trust

Article

Sinh-arcsinh distributions

M. C. Jones

Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K. m.c.jones{at}open.ac.uk

Arthur Pewsey

Department of Mathematics, Escuela Politécnica, University of Extremadura, Avenida de la Universidad s/n, 10071 Cáceres, Spain apewsey{at}unex.es

Received for publication 1 August 2008. Revision received 1 April 2009.
  Abstract

We introduce the sinh-arcsinh transformation and hence, by applying it to a generating distribution with no parameters other than location and scale, usually the normal, a new family of sinh-arcsinh distributions. This four-parameter family has 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 are superior to the state-of-the-art in normality testing because of the range of alternatives against which they are very powerful. Likelihood ratio tests of symmetry are also available and are very successful. Three-parameter symmetric and asymmetric subfamilies of the full family are also of interest. Heavy-tailed symmetric sinh-arcsinh distributions behave like Johnson SU distributions, while their light-tailed counterparts behave like 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 reparameterization. Illustrative examples are given. A multivariate version is considered. Options and extensions are discussed.

Key Words: Heavy tail • Johnson’s SUdistribution • Light tail • Sinh-normal distribution • Skew-normal distribution • Skewness • Transformation


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