Order statistic

Question

The random variables $X_1, X_2, \ldots , X_n, Y_1, Y_2, \ldots , Y_n$ are independent and $U(0, a)$-distributed. How to determine the distribution of $$Z_n = n\log\frac {\max\{X_{(n)}, Y_{(n)}\}} {\min\{X_{(n)}, Y_{(n)}\}}$$

Answer 1

Obviously, this can be written as $$ Z_n = |A_n - B_n|, $$ where $A_n = n \log X_{(n)}$, $B_n = n\log Y_{(n)}$. Notice that $A_n$ and $B_n$ are iid, so in particular $A_n - B_n$ has a symmetric distribution. Thus, the solution can be done in such steps:

1. Identify the distribution of $A_n$.

2. Using the convolution formula or otherwise identify the pdf $f_n$ of $A_n - B_n$.

3. Using the symmetry, the pdf of $Z_n$ is $g_n(x) = 2 f_n(x) \mathbf{1}_{[0,+\infty)}(x)$.