# Estimating a joint distribution from observed max and min samples

Suppose that you have jointly distributed $N$ (~100) random variables, $\{X_1,\ldots,X_N\}$, and this distribution is unknown to you. However you do know that their sum is zero by construction. Having $L$ (~3000) observations of $\max(X_1,\ldots,X_N)$ and $\min(X_1,\ldots,X_N)$ each, how can you make a statement about the joint distribution of $\{X_1,\ldots,X_N\}$?

I gave it a shot by assuming that $\{X_1,\ldots,X_N\}$ have a joint normal distribution but when I looked at the QQ-plot of the observed samples and the simulated ones, I wasn't really convinced that the underlying distribution was normal.

Even though I have no evidence that supports this hypothesis, I think I can assume that the marginal distributions of $X_1,\ldots,X_{N}$ are identical. From this assumption it follows that the mean of $X_i$ is zero for all i running from 1 to N. Furthermore, for the covariance matrix we can write
$\sigma_{ij} = \sigma^2$ whenever $i = j$
$\sigma_{ij} = -\frac{\sigma^2}{N-1}$ whenever $i \neq j$

Here $\sigma_{ij}$ is the $ij$th element of the covariance matrix and $\sigma^2$ is the variance of $X_i$ for all $i$ running from 1 to N.

• First of all, if the sum of the variables is constant, namely zero in this case, then at least one of the variables is not random， that is to say at least one of the variables is not independent.For the observations, I assume for each i.i.d observations, you only know the largest and smallest number $X_max, X_min$ Jul 9 '14 at 10:24
• For each independent variable, do they have the same distribution? If not, do they have the same mean or variance? if there are two many dependent variables and the correlations are random, the overall joint distribution, I believe, will be very complex. Jul 9 '14 at 10:30
• Extreme-value theory suggests that even a huge number of observations of the min and max of $100$ variables will reveal little about their distribution. Absent strong, quantitative assumptions, there's no hope of making extensive inferences about the full multivariate distribution based on such a dataset. As an example, let $X_1\sim$ Uniform$(-100,-99)$, $X_{100}\sim$ Uniform$(99,100)$, and let the $X_2,\ldots,X_{99}$ have any multivariate distribution $F$ supported on $(-99,99)^{98}$. These data give information only about the trivariate distribution of $(X_1,X_2+\cdots+X_{99},X_{100})$.
– whuber
Jul 9 '14 at 21:32