# Transforming a multivariate normal sample using the sinh-arcsinh transform

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.

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 $$Z$$.