The Annals of Statistics

Chernoff index for Cox test of separate parametric families

Xiaoou Li, Jingchen Liu, and Zhiliang Ying

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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


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References

  • Adler, R. J., Blanchet, J. H. and Liu, J. (2012). Efficient Monte Carlo for high excursions of Gaussian random fields. Ann. Appl. Probab. 22 1167–1214.
  • Andresen, A. and Spokoiny, V. (2014). Critical dimension in profile semiparametric estimation. Electron. J. Stat. 8 3077–3125.
  • Arcones, M. A. (2006). Large deviations for M-estimators. Ann. Inst. Statist. Math. 58 21–52.
  • Bahadur, R. R. (1960). Stochastic comparison of tests. Ann. Math. Stat. 31 276–295.
  • Bahadur, R. R. (1967). Rates of convergence of estimates and test statistics. Ann. Math. Stat. 38 303–324.
  • Berrington de González, A. and Cox, D. R. (2007). Interpretation of interaction: A review. Ann. Appl. Stat. 1 371–385.
  • Braganca Pereira, B. (2005). Separate families of hypotheses. In Encyclopedia of Biostatistics.
  • Chernoff, H. (1952). A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Ann. Math. Stat. 23 493–507.
  • Cox, D. R. (1961). Tests of separate families of hypotheses. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 105–123. Univ. California Press, Berkeley, CA.
  • Cox, D. R. (1962). Further results on tests of separate families of hypotheses. J. Roy. Statist. Soc. Ser. B 24 406–424.
  • Cox, D. R. (2013). A return to an old paper: ‘Tests of separate families of hypotheses’. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 207–215.
  • Davidson, R. and MacKinnon, J. G. (1981). Several tests for model specification in the presence of alternative hypotheses. Econometrica 49 781–793.
  • Fine, J. P. (2002). Comparing nonnested Cox models. Biometrika 89 635–647.
  • Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. In Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. I: Statistics 221–233. Univ. California Press, Berkeley, CA.
  • Kallenberg, W. C. M. (1983). Intermediate efficiency, theory and examples. Ann. Statist. 11 170–182.
  • Li, X. and Liu, J. (2015). Rare-event simulation and efficient discretization for the supremum of Gaussian random fields. Adv. in Appl. Probab. 47 787–816.
  • Li, X., Liu, J. and Ying, Z. (2017). Supplement to “Chernoff Index for Cox Test of Separate Parametric Families.” DOI:10.1214/16-AOS1532SUPP.
  • Liu, J., Lu, J. and Zhou, X. (2015). Efficient rare event simulation for failure problems in random media. SIAM J. Sci. Comput. 37 A609–A624.
  • Liu, J. and Xu, G. (2012). Some asymptotic results of Gaussian random fields with varying mean functions and the associated processes. Ann. Statist. 40 262–293.
  • Liu, J. and Xu, G. (2014a). On the conditional distributions and the efficient simulations of exponential integrals of Gaussian random fields. Ann. Appl. Probab. 24 1691–1738.
  • Liu, J. and Xu, G. (2014b). Efficient simulations for the exponential integrals of Hölder continuous Gaussian random fields. ACM Trans. Model. Comput. Simul. 24 Art. 9, 24.
  • McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models. Chapman & Hall, London.
  • Pesaran, M. H. (1974). On the general problem of model selection. Rev. Econ. Stud. 41 153–171.
  • Pesaran, M. H. (1984). Asymptotic power comparisons of tests of separate parametric families by Bahadur’s approach. Biometrika 71 245–252.
  • Pesaran, M. H. and Deaton, A. S. (1978). Testing non-nested nonlinear regression models. Econometrica 46 677–694.
  • Rukhin, A. L. (1993). Bahadur efficiency of tests of separate hypotheses and adaptive test statistics. J. Amer. Statist. Assoc. 88 161–165.
  • Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464.
  • Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics. Wiley, New York.
  • Spokoiny, V. (2012). Parametric estimation. Finite sample theory. Ann. Statist. 40 2877–2909.
  • Vuong, Q. H. (1989). Likelihood ratio tests for model selection and nonnested hypotheses. Econometrica 57 307–333.
  • White, H. (1982a). Maximum likelihood estimation of misspecified models. Econometrica 50 1–25.
  • White, H. (1982b). Regularity conditions for Cox’s test of nonnested hypothesis. J. Econometrics 19 301–318.

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.