Chernoff index for cox test of separate parametric families

Xiaoou Li, Jingchen Liu, Zhiliang Ying

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2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1-29
Number of pages29
JournalAnnals of Statistics
Issue number1
StatePublished - Feb 2018

Bibliographical note

Publisher Copyright:
© Institute of Mathematical Statistics, 2018.


  • Asymptotic relative efficiency
  • Generalized likelihood ratio
  • Generalized linear models
  • Large deviation
  • Model selection
  • Nonnested hypotheses
  • Variable selection


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