I have a sample of n iid draws from a distribution (lognormal in my case, but I would like to do it with a general distribution). I focus on the ordered statistics $X_{(1)}, X_{(2)}, ..., X_{(n)}$.
I would like to compute: $Pr(X_{(n-1)} \leq \alpha X_{(n)} + \beta)$, where $\alpha \in (0, 1)$ and $\beta$ a constant ($>0$) (eventually skip the constant I just wanted to make it more general).
So far I have that this is worth:
$\int_0^{+\infty} Pr(\alpha x \leq X_{(n-1)} \leq x) \cdot Pr(X_{(n)} = x) dx + \int_{-\infty}^{0} Pr(x \leq X_{(n-1)} \leq \alpha x) \cdot Pr(X_{(n)} = x) dx$
(remark that we can focus on distributions with positive support for what I want to do, so eventually skip the second term).
I know the formulas for the density/cdf of order statistics taken independently ( cf these notes), so I can compute this via numerical integration.
But first, I am not sure this is correct, second, I would prefer a closed-form formula as I think there should be one.
Is there any way to derive one easily?