# Multivariate logistic distribution

The normal distribution can be generalized into the multivariate normal distribution.

Can the logistic distribution also be generalized into a similar multivariate distribution? Is there a multivariate generalization of the logistic distribution which depends on the covariance matrix $\Sigma$, similar to the multivariate normal distribution? The multivariate distribution should be such that its marginals are univariate logistic distributions.

• I think that many generalizations have been proposed, see this paper.
– Anon
Feb 12, 2015 at 17:01
• Marc Nerlove is talking about multivariate logistic models in Univariate and Multivariate Log-linear/Logistic Models (1973).
– Aleh
Nov 19, 2018 at 13:35
• The best source for the multivariate logistic distribution which I know is Fang, K.-T., Xu, J.-L.: A class of multivariate distributions including the multivariate logistic. Journal of Mathematical Research and Exposition 9, 91–98 (1989) Can found at citeseerx.ist.psu.edu/viewdoc/… Jun 17, 2019 at 16:55

Using copulas you can create a multivariate distribution generalized from any univariate distribution, so yes it is possible to find a multivariate distribution with all the marginal distributions equal to logistic distributions, however it will probably not be a simple function of a covariance matrix, that relationship is pretty unique to the normal distribution.

• In (Balakrishnan '92) the following counter argument is given: for $Z_i = UV_i, \hspace{1cm} i = 1, 2, \dots , k$: "in order to have $Z$ standard logistic and thus symmetric about zero, it is necessary that the common distribution of the $V_i$'s be symmetric about zero. But it then readily follows that $\text{cov}(Z_i,Z_j)=0$". But maybe I misunderstood what he meant there. Feb 25, 2014 at 22:30
• How is that a counter-argument? The situation $Z_i=UV_i$ is not general: it posits a particular structure on the $Z_i$. Greg Snow is right that copulas can effectively be used to construct marginal-logistic multivariate distributions. Moreover, paths through the space of copulas can be used to parameterize families of such distributions having correlations varying among the extremes that are mathematically possible (the Fréchet–Hoeffding bounds). The only aspect still up in the air is whether parameters exist that are a "simple function of a covariance matrix."
– whuber
Feb 25, 2014 at 22:39
• In that case I certainly misunderstood something. However, I don't think it will be very fruitful to try to understand copulas and pursue finding the right one. It seems to be a difficult subject and I have limited time. Even though the multivariate normal distribution isn't the right one, I think I should stick to using it for the moment. Feb 26, 2014 at 0:41
• I wanted a logistic distribution because in the univariate case with parameters $\mu=0, s=1$ a transformation of the distribution using the logistic sigmoid (performing a 'change of variable') ends up in a uniform distribution. When using a normal distribution the most we can attain is a post-transformation distribution which looks kind of like a straight line, but isn't perfectly uniform. However, since I am fitting the distribution to data which can follow many distributions instead of really following a normal distribution, it is not that bad to work with the 'wrong' model. Feb 26, 2014 at 0:47

Yes. In fact, the multivariate normal and logistic distributions are members of the more general family of elliptically-contoured distributions, which can be derived from their univariate counterparts.

Both the univariate and multivariate normal distributions share the same probability density generator, which is proportional to $$g(u) = exp(-u/2)$$ That is, if $$u =\big(\dfrac{x-\mu}{\sigma}\big)^2$$ we have the normal distribution and if $$u = (x-\mu)'\Sigma^{-1}(x-\mu)$$ we have the multivariate normal distribution.

The same story goes for the univariate and multivariate logistic distributions, which share the same probability density generator, which is proportional to $$g(u) = \dfrac{exp(-u)}{(1+exp(-u))^2}.$$ If $$u =\big(\dfrac{x-\mu}{\sigma}\big)^2$$ we have the logistic distribution and if $$u = (x-\mu)'\Sigma^{-1}(x-\mu)$$ we have the multivariate logistic distribution.

Furthermore, it is well known that the parent and marginal distributions of any elliptically contoured distributions share the same type of distribution.

Most multivariate distributions discussed there contains at most two parameters which in some cases govern the covariance between the variables (besides the parameters $\mu_i$ and $\sigma_i$ for the mean and standard deviation in each variable $i$). No single distribution is given which uses (as much parameters as) the covariance matrix $\Sigma$.