# Transformation to reduce standard deviation without changing median

Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points.

Suppose that I was to create an adjusted $X_t$ called $X_t^a$ where the mean, median, skew and kurtosis remain approximately the same as the original series, and the only main difference is that $\sigma (X_t) = 2*\sigma (X_t^a)$.

What's an appropriate transformation? I've tried:

$$X_t^a := (X_t - E[X_t])*0.5\frac{\sigma(X_t)}{\sigma(X_t)} + E[X_t] \\ = Z*0.5\sigma(X_t) + E[X_t] \sim N(E[X_t],(0.5\sigma(X_t))^2)$$ $$Z \sim N(0,1)$$

but this severely reduces the positive median towards zero which is not what I want (I want to keep $\text{median}(X_t)$ approximately equal to $\text{median}(X_t^a)$).

## 2 Answers

It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved. Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be meaningful anyway (the distortions needed to mantain the statistics will do more harm than good to the empirical PDF).

Sorry if I suggest to challenge the requirements: why do you want both mean and median preserved? These statistics do not have much intrinsic value other than providing estimates of location (but not the same location though: even with full PDF knowledge they´d be different! yet one might still convert one of them to the other kind), forcing such a relationship between the two sounds like a recipe for trouble. An alternative would be to define a location statistic as some average of the two and mantain that constant in a simple linear transform. Or even better using a more robust location statistic in place of both.

As Quartz says it is possible to make non-linear transformations taking into account skew and kurtosis, but this is mostly is limited to univariate processes (one approach for a t distribution is to match moments). For multivariate processes, it is considerably more difficult. A more general solution is to rely on Entropy Pooling. You could take views on both the median and the variance of the process in a univariate or multivariate setting. Be careful to check the effective number of scenarios afterward in case you have inadvertently taken an extreme view (unlikely if fixing the median and reducing the standard deviation, but possible if making the standard deviation sufficiently large).