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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 $P(x)$?

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