CHERNOFF INDEX FOR COX TEST OF SEPARATE PARAMETRIC FAMILIES

• 成果类型: Article
• 署名作者: Li, Xiaoou; Liu, Jingchen; Ying, Zhiliang
• 署名单位: University of Minnesota System; University of Minnesota Twin Cities; Columbia University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/16-AOS1532
• 发表日期: 2018
• 页码: 1-29
• 关键词: gaussian random-fields; rare-event simulation; non-nested hypotheses; model selection; TEST STATISTICS; EFFICIENCY; dimension

摘要

The asymptotic efficiency of a generalized likelihood ratio test proposed by Cox is studied under the large deviations framework for error probabilities developed by Chernoff. In particular, two separate parametric families of hypotheses are considered [In Proc. 4th Berkeley Sympos. Math. Statist. and Prob. (1961) 105-123; J. Roy. Statist. Soc. Ser. B 24 (1962) 406-424]. The significance level is set such that the maximal type I and type II error probabilities for the generalized likelihood ratio test decay exponentially fast with the same rate. We derive the analytic form of such a rate that is also known as the Chernoff index [Ann. Math. Stat. 23 (1952) 493-507], a relative efficiency measure when there is no preference between the null and the alternative hypotheses. We further extend the analysis to approximate error probabilities when the two families are not completely separated. Discussions are provided concerning the implications of the present result on model selection.