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If I recall correctly, we can add and subtract variance if variables are independent, and have a mean of 0.

I have two distributions that are summed up: (a) one with high variance, low skewness and low kurtosis, and (b) one with lower variance, higher skewness and higher kurtosis.

I'd like to have a mathematically sound way of subtracting out portions of distribution (b). As the proportion of (a) to (b) increases, the standard deviation is increasing, while skewness and kurtosis are decreasing. Variance is the easy one, but how do I do this for skewness and kurtosis?

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  • $\begingroup$ I would try characteristics functions $\endgroup$ – Aksakal Oct 6 '17 at 15:51
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    $\begingroup$ Could you clarify what "add and subtract" variance means and does? What about "subtracting out portions" of a distribution--how is that supposed to work, exactly?? $\endgroup$ – whuber Oct 6 '17 at 16:42
  • $\begingroup$ The context is the expected return distribution of a stock -- the market expects a certain amount of variance in the price every day. If the market expects 2 variance to be realized every day, then the monthly variance would be 44. Every day, we can subtract out the variance that people expected. When I talk about the sum of two distributions, let's pretend I'm talking about the expected return over the next month. Distribution (b) describes the behavior of the stock on any given day, while distribution (a) describes the behavior of the stock on an event. $\endgroup$ – OGC Oct 6 '17 at 17:45
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The quantities (invariants) that add for independent variables are the cumulants: see https://en.wikipedia.org/wiki/Cumulant

Skewness, as defined in https://en.wikipedia.org/wiki/Skewness is not a cumulant, so is not additive. But it can be expressed using cumulants, so you can first add up the cumulants and then calculate skewness. Looking up the formulas in wiki above, you will see that if the variance is a constant, then, in that case only, the skewness will be additive.

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