Solved – Normalizing the joint probability density

copuladensity functionjoint distributionprobability

I computed the kernel estimators for the copula density for two random variables using:

library(kdecopula)

kde.fit <- kdecop(u)

As the values of density can be greater than one I was wondering if I can normalized values by the maximum magnitude and call it normalized joint probability density?

Here is the bivariate copula density and the normalized bivariate copula density

Bivarite copula density:
enter image description here

Normalized bivarite copula density:

enter image description here

Thanks in advance for any helps.

Best Answer

In literature, normalization means integrating to $1$, not having a max value equal to $1$. So, joint or univariate densities are already normalized. For the nomenclature, for the function you have, I think max-normalized joint density would be a better name for it.

However, what you do is just scaling your joint PDF so that it hits $1$ at its maximum. Since both in your new function and the original density, these values can't be associated with probabilities, I see no use in doing so. It's not as much different as than multiplying your density with e.g. $5$.

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