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For a given random variable $X$ with mean $\mu_{\mathrm{old}}$ and standard deviation $\sigma_{\mathrm{old}}$ I would like to perform a transformation $g$ to obtain a new random variable $Y := g(X)$ which has a desired mean and standard deviation $\mu_{\mathrm{new}}$ and $\sigma_{\mathrm{new}}$ respectively without (if possible) changing other characteristics of the distribution (higher moments). A simple calculation shows that the transformation $g$ has the form

$$Y = g(X) := \frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}X-\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}$$

This fulfills the condition above since

$$E[Y] = \frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}E[X]-\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}=\frac{\sigma_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}}-\frac{\mu_{\mathrm{new}}}{\sigma_{\mathrm{old}}}\mu_{\mathrm{old}} + \mu_{\mathrm{new}}=\mu_{\mathrm{new}}$$

and

$$\operatorname{Var}(Y) = \frac{\sigma^2_{\mathrm{new}}}{\sigma^2_{\mathrm{old}}}\operatorname{Var}(X)=\sigma^2_{\mathrm{new}}.$$

Does this normalization and scaling change other moments? I did some simulation in R and it seems no to be the case, at least for kurtosis and skewness. What about higher moments? Are there any reference for this?

Davide Giraudo
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1 Answers1

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Skewness and kurtosis are not actually moments of the random variable itself (either raw or central). They're central moments divided by the corresponding power of $\sigma$, to make them unit-free.

e.g. The third raw moment $\mu_3' = E(X^3)$ is affected by both changes of location and scale, the third central moment is $\mu_3=E[(X-\mu)^3]$ is not affected by changes of location (since that's removed when $\mu$ is subtracted) but is affected by change of scale. The skewness, by contrast, is $\mu_3/\sigma^3$; it's effectively the third moment of a standardized version of the original random variable.

Let $\mu_k$ be the $k$-th central moment of $X$ and let $Y=c(X-\mu_X)$. Then $E(Y^k)=E([c(X-\mu_X)]^k) = E(c^k[(X-\mu_X)]^k) = c^kE([(X-\mu_X)]^k)=c^k\mu_k$

So while $\mu_3$ is affected by change of scale, $\mu_3/\sigma^3$ (third-moment skewness) is not.

Similar comments apply to raw fourth moments, central fourth moments and kurtosis and the same ideas extend to higher order moments - raw moments are affected by both kinds of change, central moments by scale changes only, and standardized central moments by neither.

Glen_b
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    +1. In terms often used, skewness and kurtosis measure aspects of distribution shape but shape is invariant under changes of location or scale. – Nick Cox Jul 06 '16 at 13:18
  • @Glen_b or Nick, do you have any references on this analysis of moments. Most stats texts I have seen never do a deep analysis of moments beyond 2. I am working on a problem where a bit more precision is needed in understanding how scaling affects the skewness of a distribution. – krishnab Mar 06 '19 at 04:18
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    The line of algebra in the answer ($E(Y^k)=E([c(X-\mu_X)]^k) = E(c^k[(X-\mu_X)]^k)$ $= c^kE([X-\mu_X]^k)=c^k\mu_k$) is pretty much the whole shebang; It follows from the fact that $(ab)^n = a^n b^n$, and the linearity of expectation. Some elementary books may have exercises on it but presumably nobody's going to devote more than a few lines to stating it. The skewness will have a factor of the cube of the scale in both numerator and denominator, which cancel, so scale doesn't affect skewness at all; it is unitless. – Glen_b Mar 06 '19 at 05:43
  • What if instead of scaling, we take the logarithm of $X$ ? How does skewness and kurtosis change then ? – Our Jul 19 '19 at 19:55
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    That's a whole new question... – Glen_b Jul 20 '19 at 01:55