The Annals of Statistics

Chernoff index for Cox test of separate parametric families

Xiaoou Li, Jingchen Liu, and Zhiliang Ying

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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.

Article information

Ann. Statist., Volume 46, Number 1 (2018), 1-29.

Received: January 2016
Revised: November 2016
First available in Project Euclid: 22 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing
Secondary: 62J12: Generalized linear models 62F12: Asymptotic properties of estimators

Asymptotic relative efficiency generalized likelihood ratio generalized linear models large deviation model selection nonnested hypotheses variable selection


Li, Xiaoou; Liu, Jingchen; Ying, Zhiliang. Chernoff index for Cox test of separate parametric families. Ann. Statist. 46 (2018), no. 1, 1--29. doi:10.1214/16-AOS1532.

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Supplemental materials

  • Supplement to “Chernoff index for Cox test of separate parametric families”. In the supplement [Li, Liu and Ying (2017)], we present proofs of Corollary 6, Lemmas 17, 18 19, 20 and 21.