Concentration inequalities, large and moderate deviations for self-normalized empirical processes

• 成果类型: Article
• 署名作者: Bercu, B; Gassiat, E; Rio, E
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1576-1604
• 关键词: limit-theorems; MODEL

摘要

We consider the supremum W-n of self-normalized empirical processes indexed by unbounded classes of functions F. Such variables are of interest in various statistical applications, for example, the likelihood ratio tests of contamination. Using the Herbst method, we prove an exponential concentration inequality for W-n under a second moment assumption on the envelope function of F. This inequality is applied to obtain moderate deviations for W-n. We also provide large deviations results for some unbounded parametric classes F.