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I was reading through Meucci's Risk and Asset Allocation (2005), when I happened upon the concept of a "density generator", which I have not been able to find good explanations for anywhere online, and would appreciate some help in understanding.

From pg. 116 of the book, the density generator, $g_N$, is introduced,

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and later, on pg. 214, the normal distribution's generator is shown as

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Based on searches online, it seems that density generators have to do with copulas, and represent a method of specifying tail dependence within elliptical distributions, which normally do not feature such dependence (?), as described in this book.

While I grasp the basics of copula theory, some help in understanding this concept would be much appreciated.

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An elliptical distribution is affinely equivalent to a spherically symmetric distribution. It therefore suffices to specify the radial distribution. By convention, it is given in terms of the distribution of the squared radius. When that is continuous, its density is the generator.


This means that when an $n$-dimensional random variable $X$ has an elliptical distribution, there is a center $\mu\in\mathbb{R}^n$ and an invertible matrix $\Lambda$ such that the distribution of

$$Z = \Lambda^{-1}(X-\mu)$$

is the same as the distribution of $QZ$ for any orthogonal matrix $Q.$ Equivalently, the distribution of $Z$ determines and is determined by the distribution of its squared Euclidean length

$$|Z|^2 = Z^\prime Z = (X-\mu)^\prime \Lambda^{-1^\prime} \,\Lambda^{-1}(X-\mu) = (X-\mu)^\prime \Sigma^{-1} (X-\mu)$$

where $\Sigma=\Lambda \Lambda^\prime$ is (clearly) symmetric and positive definite.

When such a distribution is continuous it has a multivariate density $f,$ which therefore can be written in the form

$$f(x) = k\,g\left((x-\mu)^\prime \Sigma^{-1} (x-\mu)\right)$$

with $k$ the constant that makes $f$ integrate to unity. According to Wikipedia, $g$ is a generator of the distribution of $X.$

For example, because the standard elliptical Normal distribution has a density proportional to $$\exp(-(z_1^2+z_2^2+\cdots+z_n^2)/2) = \exp\left(-|z|^2/2\right) = g(|z|^2),$$ its generator is proportional to

$$g(z) = e^{-z/2}, z \ge 0;\ g(z) = 0, z \lt 0.$$

(It's always understood that $g$ only needs to be defined for $z\ge 0.$)

Note that $g$ is not the density of $|Z|^2$ nor even of $|Z|$ itself: that's because in spherical coordinates in $\mathbb{R}^n,$ $g$ must be multiplied by $|z|^{n/2-1}$ to obtain the density of $|Z|^2.$ For instance, the distribution of the squared spherical Normal radius has a density proportional to $z^{n/2-1}e^{-z/2},$ a Chi-squared distribution.

Because it can help to have more than one example of something, here are a few other generators.

The multivariate Student t distribution

For the distribution with $\nu$ degrees of freedom, a generator is

$$g_\nu(z) = \frac{1}{\left(1 + z\right)^{(\nu+n)/2}}.$$

The multivariate uniform spherical distribution

A generator is

$$g(z) = 1, 0 \le z\le 1;\ g(z) = 0\text{ otherwise.}$$


This concept of "generator" has little to do with copulas per se and almost nothing to do with tail behavior, either. ($g$ determines its tail behavior, of course; but there are many possible functions $g$ associated with any radial tail behavior.)

The account I gave of the Mahalanobis distance applies equally well to understanding any multivariate distribution, showing how $\mu$ merely establishes the origin of a coordinate system and $\Sigma$ determines a set of orthogonal coordinate axes in which the multivariate distribution of $X$ has a particularly simple form. What makes the distribution "elliptical" is that the resulting form is especially simple, depending only on the (Mahalanobis) distance from the origin.

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Generator functions of elliptical distributions is a function that characterises the tail dependency in each function. For example, normal distribution does not has a tail dependency, while t-student is symmetric tail function. A copula is just a multivariate distribution function with standard uniform margins. Normal copula function also cannot deal with tail dependency.

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  • $\begingroup$ Can density generators be used for other distributions as well, then? Can any arbitrary, continuous distribution have a generator attached to it, that would allow it to feature tail dependence, where no existed before? $\endgroup$ – Coolio2654 Jul 9 at 20:22
  • $\begingroup$ A generator function always exists. Each distribution has its own generator function. If you change the generator function of Gaussian, how can it become a Gaussian distribution?! For example, if we use the generator function of Gaussian with T-student distribution, do you think it will be still a Gaussian distribution?! Each Archimedian copula function, for example, has its own generator function. There are some conditions that make the generator function valid or not! $\endgroup$ – Mary Jul 10 at 6:01

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