# Standardize first four moments: match sample moments with population moments

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 $$4-\delta$$.

• Why not use your second equation with $\epsilon=0$ and $\delta=1$? – Dave May 16 '20 at 12:18
• @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. – Alex May 16 '20 at 14:37
• You cannot achieve this without some kind of nonlinear transformation. Please explain what family of transformations you are willing to consider. – whuber May 16 '20 at 21:48
• @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? – Alex May 16 '20 at 22:05
• 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. – whuber May 16 '20 at 22:08