In My field , I reached to this problem.
Assumptions:
Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ such pairs.
Now the question is :
$$P=P_r\left(\frac{\max(x_i)}{\max(\hat{x}_i)}>K\right)$$
My Answer:
I started with induce concomitant theorem (concomitant order static) as below: I am ordering in ascending: $x_1<x_2<\cdots<x_N$
Therefore, we ave: $$ P=1-\Pr(\max(x_i)<K\max(\hat{x}_i))$$ $$ \Pr(\max(x_i)<\max(\hat{x}_i))=P_r(x_1<\hat{x}_N K, x_2<\hat{x}_N K,\ldots,x_N<\hat{x}_N K)$$
but, it seems to be very complex integration! (exhaustive one). i mean, we must let $x_n=t$ which is translating to $N-1$ integrals based on $t$. Ultimately, we must integrate the resulted expression with respect to $x_n$. I will be happy to have your comments, hints or suggestion.