# How to fully estimate a probability density from only a sample of minimum values?

We are given a sample $$\{ z_i \}$$, $$i=1,2,\ldots,N$$, such that each value $$z_i$$ corresponds to the minimum of $$n$$ random variables $$x$$, i.e., $$z = \min \{ x_1, x_2,\ldots,x_n \}$$.

By means of elementary order statistics and the empirical distribution function $$F_z$$ of $$z$$, we can do $$F_x = 1-(1-F_z)^{1/n}$$ to obtain the CDF of $$x$$ (see https://stats.stackexchange.com/q/10072). However, since $$z$$ are minimum values, the sample $$\{ z_i \}$$ can hardly tell us anything about the right tail of the density, $$P(x \rightarrow\infty)$$.

The question is: how can we improve the elementary method stated above in order to be able to say something about the right tail of $$P(x)$$?