Once more on stability estimation in the problem of reconstructing the additive type of a distribution

L. B. Klebanov

Zapiski Nauchnykh Seminarov POMI, Studies in mathematical statistics. Part 3, Tome 87 (1979), pp. 74-78

Résumé

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).