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

Majboroda, R. E.

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 \(\{\xi_j\}\).

Reviewer: O.V.Sugakova (Kyïv)

MSC:

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

Keywords:

extreme points; mixture models; varying concentrations; convergence rate