# Impact of having negative Box-Cox lambdas on a Tau correlation matrix

I'm trying to make the marginal distributions of my data easier to fit by first applying a Box-Cox transformation and then standardizing the transformed data. My concern is, as this is a multivariate Monte Carlo simulation, that a negative Box-Cox lambda would throw off my Tau correlation matrix. For now, to not upset rankings, I've enforced a non-negativity constraint on Lambda -- which limits some of my time series. Is there a way around this?

TL;DR: How can I allow for a negative Box-Cox lambda without disrupting my Tau correlation matrix?

Thank you for your time and kind assistance!

• In what sense would a negative Box-Cox parameter "throw off" or "disrupt" your correlation matrix? Remember the definition of the Box-Cox transformation guarantees it will be monotonic even for negative parameters, so if $\tau$ is based on ranks (such as Kendall's tau), the correlation won't change at all. – whuber Aug 28 '20 at 18:26
• If I have a negative lambda, wouldn't I get an inverse of the original data, in which case the larger the original values, the lower the transformations, which would reverse direction of the correlation? Thanks! – CasusBelli Aug 28 '20 at 19:04
• You are thinking of the power transformation $x \to x^\lambda.$ The Box-Cox transformation is $x \to \int_1^x y^{\lambda-1}\mathrm{d}y = (x^\lambda-1)/\lambda$ (the equality holds only for $\lambda\ne 0$). See stats.stackexchange.com/a/467525/919 for an illustrated discussion of what this accomplishes. Among other things, division by $\lambda$ prevents the correlation from being reversed (negated). – whuber Aug 28 '20 at 19:43