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Is there a name for this type of transformation:

\begin{equation} sgn(x) * |x|^p \end{equation} where $p$ is an arbitrary number (e.g., 1/2 and 1/3 for square and cube roots, respectively) and $sgn$ is the signum function?

Here something similar is suggested for a transformation using an arbitrary $p$.

Are these valid transformation prior to regression etc. (to bring heavy-tailed distributions closer to normal)? They seem to offer less "compression" over a wide range of numbers than the inverse hyperbolic sine does (which is often recommended for data sets with negative values).

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    $\begingroup$ It is natural, and tempting, to try something like this with data that are differences of skewed data. It rarely works (and has extremely poor properties for negative $p.$) If this question was motivated by your analysis of a dataset, why not ask directly about the original issue? $\endgroup$ – whuber Oct 7 at 21:58
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    $\begingroup$ I don't think it can even be discussed usefully unless $p > 0$. The case of the cube root seems the commonest application, which I have found helpful for visualization and analysis of glacier terminus change e.g. Nature 500, 563–566 (2013).and profile and plan curvature in geomorphometry. $\endgroup$ – Nick Cox Oct 7 at 22:06
  • $\begingroup$ @whuber when you say it rarely "works"/has poor properties, what type of properties do you speak of? If these transformations make the data more normal to conform to e.g., regression methods, are there more that we should consider? And yes I did not consider the $p<0$ case but for $p>0$ it could be fine? $\endgroup$ – hatmatrix Oct 9 at 10:35
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    $\begingroup$ The poorest property is that it is discontinuous and unbounded at the origin for non-positive $p.$ Regression methods rarely require Normality: one seeks approximate symmetry of distribution. Often, data that span zero have been computed as differences and in those cases, if it's possible, transforming the data before subtracting them is more effective and more readily interpretable. $\endgroup$ – whuber Oct 9 at 13:14
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    $\begingroup$ Often, a good model for that situation is a lognormal distribution of true concentrations plus Normal errors. Again, attempting to transform the data in some ad hoc way is usually neither effective nor insightful, because it is trying to cope with the effects of both phenomena simultaneously. $\endgroup$ – whuber Oct 11 at 14:29

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