## The Annals of Statistics

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

### Chernoff index for Cox test of separate parametric families

Xiaoou Li, Jingchen Liu, and Zhiliang Ying

#### Abstract

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

**Source**

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

**Dates**

Received: January 2016

Revised: November 2016

First available in Project Euclid: 22 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1519268422

**Digital Object Identifier**

doi:10.1214/16-AOS1532

**Mathematical Reviews number (MathSciNet)**

MR3766944

**Zentralblatt MATH identifier**

06865103

**Subjects**

Primary: 62F03: Hypothesis testing

Secondary: 62J12: Generalized linear models 62F12: Asymptotic properties of estimators

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1519268422

#### 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.Digital Object Identifier: doi:10.1214/16-AOS1532SUPPSupplemental files are immediately available to subscribers. Non-subscribers gain access to supplemental files with the purchase of the article.