Sinh-arcsinh distributions

Abstract

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.

Modeling Non-Stationary Asymmetric Lens Blur by Normal Sinh-Arcsinh Model

“…In statistics, the skew-normal distribution and normal sinh-arcsinh (NSAS) distribution are used to model Gaussian-like distributions with skewness [12], [13], [14], [15]. Skew-normal distribution is obtained by multiplying the Gaussian density and the cumulative distribution together [12].…”

“…The skew-normal distribution retains the general shape of the Gaussian function with the skewness toward one direction. The NSAS distribution has two specific parameters that control the skewness and the kurtosis of the distribution [13], [14], [15]. The two dimensional NSAS distribution has more flexible shapes than the skew-normal distribution.…”

Integrated analysis of genome-wide genetic and epigenetic association data for identification of disease mechanisms

“…27 Permutations were performed to assess the significance of the sum of such chi-square statistic values (Sst) for all the SNPs in a pathway, and the resulting null distribution of the summary statistic was approximated using the Sinh-Arcsinh (SHASH) probability distribution. 28 This model has four parameters, governing location (μ), scale (σ) and shape (ν, τ) and offers sufficient flexibility to fit a large number of empirical distributions; it can be easily fitted with the gamlss R package (http:// www.gamlss.com). 28,29 We generated 10 000 permutations of the sample case/control labels, and estimated the four parameters µ, σ, ν, τ in a SHASH model by maximizing the likelihood P(Sst k |µ,σ,ν,τ), k [1,10 000], where each Sst k was calculated as the sum statistic under a random permutation of the case control labels and k was the index of the permutation.…”

“…28 This model has four parameters, governing location (μ), scale (σ) and shape (ν, τ) and offers sufficient flexibility to fit a large number of empirical distributions; it can be easily fitted with the gamlss R package (http:// www.gamlss.com). 28,29 We generated 10 000 permutations of the sample case/control labels, and estimated the four parameters µ, σ, ν, τ in a SHASH model by maximizing the likelihood P(Sst k |µ,σ,ν,τ), k [1,10 000], where each Sst k was calculated as the sum statistic under a random permutation of the case control labels and k was the index of the permutation. The P value of the sum statistic was then obtained as P = , where Sst was the observed sum statistic of the real data and P(x|μ,σ,ν,τ) was the fitted SHASH distribution.…”

Multivariate skewing mechanisms: A unified perspective based on the transformation approach

“…There exist other ways of skewing symmetric distributions, for example by using order statistics (Jones, 2004), by applying Tukey's g-and-h transformations (Hoaglin 1986, or Field andGenton 2006 for the multivariate case) or through the sinh-arcsinh transformation (Jones and Pewsey, 2009). Each of the pre-cited skewing methods has advantages and disadvantages, but they all share a common drawback: most properties of the resulting skewed distributions strongly depend on the choice of the method, hence follow a pre-defined pattern and cannot be adapted according to the needs in certain situations.…”