## An estimate for the extremal points of a distribution from observations of a mixture with varying concentrations.(English. Ukrainian original)Zbl 0947.62035

Theory Probab. Math. Stat. 60, 131-135 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 118-122 (1999).
The paper deals with estimation of the distribution of extreme points $m_H=m_\xi=\text{ess inf}(\xi)=\sup\{x: P\{\xi<x\}=H(x)=0\}.$ If an i.i.d. sample $$\xi_1$$,…,$$\xi_N$$ of $$\xi$$ is available, then the usual estimator $$\widehat m_N=\min_j\{\xi_j\}$$ provides the convergence rate $|\widehat m_N-m_H|\leq (A_N\ln N)N^{-1} \text{a.s.}\tag{1}$ for any $$A_N\to\infty$$ as $$N\to\infty$$. The case is considered where $$m_{H_l}$$ is estimated by a sample from a mixture with varying concentrations, i.e. $$\xi_j$$, $$j=1,\dots,N$$, are independent and $P\{\xi_j<x\}=\sum_{k=1}^M w_j^k H_k(x),$ where $$w_j^k$$ are known mixture probabilities (concentrations), and $$H_k$$ are unknown distributions of the mixture components. An estimator $$\widehat m_N(A_N)$$ is proposed for which (1) holds for any $$A_N\to\infty.$$ This estimator is based on a histogram-like transform of a weighted empirical distribution function of $$\{\xi_j\}$$.

### MSC:

 62G32 Statistics of extreme values; tail inference 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation