Let $X$ be a sample from $N(0,1)$ and $m$, $v$, $s$, $k$ denote sample mean, variance, skewness and kurtosis of $X$. I want to transform the sample $X$ such that the sample moments equal the true population moments, e.g.

  • sample mean = 0
  • sample variance = 1
  • sample skewness = 0
  • sample kurtosis = 3
  • ...

Using z-scores, $\frac{X-m}{\sqrt{v}}$, I can match the first two moments perfectly.

I seek a (nonlinear) transformation which helps my sample to match further population moments.

I found online the sinh-arcsinh transformation, that is $$Z=\sinh\left((4-k)\sinh^{-1}\left(\frac{X-m}{\sqrt{v}}\right)-s\right),$$

which should result in a match of the first four sample moments with the true population moments.

However, if I compare this transformation with the plain z-scores, $\frac{X-m}{\sqrt{v}}$, then that simpler approach yields better results (sample moments match population moments more closely). How can I transform the data correctly to match the moments?


Let $Z\sim N(0,1)$. Then, $$X=\mu+\sigma\sinh\left(\frac{\sinh^{-1}\left(Z\right)+\varepsilon}{\delta}\right)$$ has mean $\mu$, variance $\sigma^2$, skewness $\varepsilon$ and kurtosis $4-\delta$.

  • $\begingroup$ Why not use your second equation with $\epsilon=0$ and $\delta=1$? $\endgroup$ – Dave May 16 '20 at 12:18
  • $\begingroup$ @Dave I thought the second equation requires standardised data as input? But the sample moments of my sample $X$ are not quite equal to 0, 1, 0 and 3. That’s the aim to get these ideal moments. $\endgroup$ – Alex May 16 '20 at 14:37
  • $\begingroup$ You cannot achieve this without some kind of nonlinear transformation. Please explain what family of transformations you are willing to consider. $\endgroup$ – whuber May 16 '20 at 21:48
  • $\begingroup$ @whuber I'm happy with any nonlinear transformation. I generated a sample $X$ of $N(0,1)$ realisations. But of course the sample moments of $X$ are not precisely equal to the population moments. So I seek some transformation $f$ such that $f(X)$ has the ''right'' first four sample moments. I thought the (nonlinear) sin-arcsin transformation may work but its success is underwhelming. Do you have better ideas? $\endgroup$ – Alex May 16 '20 at 22:05
  • 1
    $\begingroup$ Unfortunately, your question isn't sufficiently specific. You can always transform a dataset to make it extremely close to any given distribution by means of the probability integral transform, but there are myriad other ways, too. $\endgroup$ – whuber May 16 '20 at 22:08

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