Abstract:
Let $X$ and $Y$ be two random vectors with characteristic functions $\varphi(t)$ and $\psi(t)$, $t=(t_1,\dots,t_m)\in R^m$, respectively. The distance $\nu(X,Y)$ is defined by
$$
\nu(X,Y)\inf_{T>0}\max\biggl\{\sup_{|t_1|\leq T}|\varphi(t)-\psi(t)|,1/T\biggr\}.
$$
Suppose that for two distribution functions $F$ and $F_1$ of a random variable $x_i$ the corresponding distributions $H$ and $H_1$ of the maximum invariant statistic $Y=(x_2-x_1,\dots,x_n-x_1)$ of the sample
$(x_1,\dots,x_n)$ are $\varepsilon$-close in sense of $\nu(Y/H,Y/H_1)\leq\varepsilon$. Then for some $\theta\in R^1$ $$
\nu(x_1|_F, x_1+\theta|_{F_1})\leq c\cdot\max(\varepsilon^{1-(2k-1)\lambda},
\varepsilon_\lambda,1/B_{m+1}^{(\lambda)}(\varepsilon))
$$
where
$B_{m+1}^{(\lambda)}(\varepsilon)$ is defined by (2).