Suppose that $X_1,...,X_n$ are independent and identically distributed (iid) random variables with a common cdf $F_X(x)$. Similarly, suppose that $Y_1,...,Y_n$ are iid random variables with a common cdf $F_Y(x)$. ($X_i$ and $Y_i$ are also independent.) Define $Z_i = X_iY_i$ and let $\pi:\{1,...,n\}\to\{1,...,n\} $ be a permutation function such that $$Z_{\pi(1)} \le Z_{\pi(2)} \le ...\le Z_{\pi(n)}.$$ Note that $Z_{\pi(1)},...,Z_{\pi(n)}$ is the order statistics ($\pi$ function is defined by sorting the realizations of $Z_1,...,Z_n$ in increasing order).
I easily obtained the pdf of $Z_{\pi(i)}$ by using $$f_{Z_{\pi(i)}}(x) = \frac{n!}{(i-1)!(n-i)!} [F_Z(x)]^{i-1}[1-F_Z(x)]^{n-i}f_Z(x),$$ where $$f_Z(z) = \int_{-\infty}^{\infty} f_X(x)f_Y(z/x)1/|x|dx.$$
I tried to derive the cdf of $Y_{\pi(i)}$, but failed. Any ideas?
Addition: $Y_{\pi(i)}$ is not the order statistcs of $Y_i$. That is, $\pi(1)$ is the index of the smallest $Z_i$, but is not the index of the smallest $Y_i$.