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Let us say that we have sample from the multivariable normal distribution. I would like to understand how is possible to apply a transformation to this sample to produce sample that has the sinh-arcsinh distribution written in following paper: Sinh-Archsinh Distributions

My question is related to previous question: Transformation to increase kurtosis and skewness of normal r.v , except in multivariable case. In previous question there is the inverse transform which takes normal sample and gives sinh-arcsinh sample. Is there equivalent transform to the multivariable sample?

Alternative is to take the multivariable pdf of sinh-arcsinh distribution and get the sample directly from that, but I am curious if it can be done from the multivariable normal sample.

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The basic idea is described in the paper quoted Sinh-Archsinh Distributions and boils down to the component-by-component application of the inverse transform you mention:

  1. generate $d$-dimensional Multivariate Normal variables $Z\sim \mathcal{N}(0,R)$, where R is a $d\times d$ correlation matrix (using e.g. method described here).

  2. chose $d$ sets of skew and kurtosis parameters $(\epsilon_i,\delta_i)_{i=1,\cdots,d}$, with $\epsilon_i\in \mathbb{R}, \delta_i \in \mathbb{R}_0^{+}$

  3. define your multivariate sin-asinh component-by component as follows : $X_{\epsilon_i,\delta_i}\equiv \sinh(\frac{\text{arcsinh}(Z_i)-\epsilon_i}{\delta_i}), \text{ for } i=1,\cdots,d$ , where $Z_i$ is the $i-$th component of your multivariate normal $Z$.

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