Abstract:
Let $U_n(1),\dots,U_n(n)$ be the variational series of a simple random sample of size $n$ from the uniform distribution on [0, 1].
In this paper, the asymptotical distribution (as $n\to\infty$) of statistic
$$
\xi_n=\frac{1}{2}\sum_{j=1}^{n+1}a\biggl(\frac{j}{n+1}\biggr)
\biggl|\varphi_n(U_n(j))-\varphi_n(U_n(j-1))-\frac{1}{n+1}\biggr|
$$
is derived, where $a(u)$, $0\le u\le 1$, is a weight function,
$$
\varphi_n(u)=u+\frac{1}{\sqrt{n+1}}\int_0^u b_n(x)\,dx,\qquad\int_0^u b_n(x)\,dx=O(1).
$$
The result obtained is used to construct a goodness-of-fit test.
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