Annals of Statistics

On asymptotically optimal tests under loss of identifiability in semiparametric models

Rui Song, Michael R. Kosorok, and Jason P. Fine

Full-text: Open access


We consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models, where regularity conditions for profile likelihood theory fail. Exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model. These results extend existing results for parametric models, which involve more restrictive assumptions on the form of the alternative than do our results. Moreover, the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate. Examples include tests of the presence of a change-point in the Cox model with current status data and tests of regression parameters in odds-rate models with right censored data. Optimal tests have not previously been studied for these scenarios. We study the asymptotic distribution of the proposed tests under the null, fixed contiguous alternatives and random contiguous alternatives. We also propose a weighted bootstrap procedure for computing the critical values of the test statistics. The optimal tests perform well in simulation studies, where they may exhibit improved power over alternative tests.

Article information

Ann. Statist., Volume 37, Number 5A (2009), 2409-2444.

First available in Project Euclid: 15 July 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62A01: Foundations and philosophical topics 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties 62C99: None of the above, but in this section

Change-point models contiguous alternative empirical processes exponential average test nonstandard testing problem odds-rate models optimal test power profile likelihood


Song, Rui; Kosorok, Michael R.; Fine, Jason P. On asymptotically optimal tests under loss of identifiability in semiparametric models. Ann. Statist. 37 (2009), no. 5A, 2409--2444. doi:10.1214/08-AOS643.

Export citation


  • Andrews, D. W. K. (1999). Estimation when a parameter is on a boundary. Econometrica 67 1341–1383.
  • Andrews, D. W. K. (2001). Testing when a parameter is on the boundary of the maintained hypothesis. Econometrica 69 683–734.
  • Andrews, D. W. K. and Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative (STMA V36 3473). Econometrica 62 1383–1414.
  • Andrews, D. W. K. and Ploberger, W. (1995). Admissibility of the likelihood ratio test when a nuisance parameter is present only under the alternative. Ann. Statist. 23 1609–1629.
  • Bagdonavičius, V. B. and Nikulin, M. S. (1999). Generalized proportional hazards model based on modified partial likelihood. Lifetime Data Anal. 5 329–350.
  • Bickel, P. J., Ritov, Y. and Stoker, T. M. (2006). Tailor-made tests for goodness of fit to semiparametric hypotheses. Ann. Statist. 34 721–741.
  • Chappell, R. (1989). Fitting bent lines to data, with applications to allometry. J. Theoret. Biol. 138 235–256.
  • Chen, H. and Chen, J. (2003). Tests for homogeneity in normal mixtures in the presence of a structural parameter. Statist. Sinica 13 351–365.
  • Chen, H., Chen, J. and Kalbfleisch, J. D. (2004). Testing for a finite mixture model with two components. J. Roy. Statist. Soc. Ser. B Stat. Methodol. 66 95–115.
  • Chernoff, H. (1954). On the distribution of the likelihood ratio. Ann. Math. Statist. 25 573–578.
  • Chernoff, H. and Lander, E. (1995). Asymptotic distribution of the likelihood ratio test that a mixture of two binomials is a single binomial. J. Statist. Plann. Inference 43 19–40.
  • Dacunha-Castelle, D. and Gassiat, E. (1999). Testing the order of a model using locally conic parametrization: Population mixtures and stationary ARMA processes. Ann. Statist. 27 1178–1209.
  • Davies, R. B. (1977). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64 247–254.
  • Davies, R. B. (1987). Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74 33–43.
  • Golub, G. H. and Van Loan, C. F. (1983). Matrix Computations. Johns Hopkins Univ. Press, Baltimore, MD.
  • Huang, J. (1996). Efficient estimation for the proportional hazards model with interval censoring. Ann. Statist. 24 540–568.
  • King, M. L. and Shively, T. S. (1993). Locally optimal testing when a nuisance parameter is present only under the alternative. Rev. Econom. Statist. 75 1–7.
  • Kosorok, M. R. (2008). Introduction to Empirical Processes and Semiparametric Inference. Springer, New York.
  • Kosorok, M. R., Lee, B. L. and Fine, J. P. (2004). Robust inference for univariate proportional hazards frailty regression models. Ann. Statist. 32 1448–1491.
  • Kosorok, M. R. and Song, R. (2007). Inference under right censoring for transformation models with a change-point based on a covariate threshold. Ann. Statist. 35 957–989.
  • Lindsay, B. G. (1995). Mixture Models: Theory, Geometry and Applications. Institute of Mathematical Statistics.
  • Liu, X. and Shao, Y. (2003). Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Statist. 31 807–832.
  • Luo, X., Turnbull, B. W. and Clark, L. C. (1997). Likelihood ratio tests for a changepoint with survival data. Biometrika 84 555–565.
  • Murphy, S. A., Rossini, A. J. and van der Vaart, A. W. (1997). Maximum likelihood estimation in the proportional odds model. J. Amer. Statist. Assoc. 92 968–976.
  • Murphy, S. A. and van der Vaart, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25 1471–1509.
  • Murphy, S. A. and van der Vaart, A. W. (1999). Observed information in semi-parametric models. Bernoulli 5 381–412.
  • Murphy, S. A. and van der Vaart, A. W. (2000). On profile likelihood (C/R: P466-485). J. Amer. Statist. Assoc. 95 449–465.
  • Parner, E. (1998). Asymptotic theory for the correlated gamma-frailty model. Ann. Statist. 26 183–214.
  • Pollard, D. (1995). Another look at differentiability in quadratic mean. Available at
  • Pons, O. (2003). Estimation in a Cox regression model with a change-point according to a threshold in a covariate. Ann. Statist. 31 442–463.
  • Scharfstein, D. O., Tsiatis, A. A. and Gilbert, P. B. (1998). Semiparametric efficient estimation in the generalized odds-rate class of regression models for right-censored time-to-event data. Lifetime Data Anal. 4 355–391.
  • Slud, E. V. and Vonta, F. (2004). Consistency of the NPML estimator in the right-censored transformation model. Scand. J. Statist. 31 21–41.
  • van der Vaart, A. W. (1996). Asymptotic Statistics. Cambridge Univ. Press, Cambridge.
  • van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
  • Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc. 54 426–482.
  • Zhu, H. T. and Zhang, H. P. (2004). Hypothesis testing in mixture regression models. J. Roy. Statist. Soc. Ser. B Stat. Methodol. 66 3–16.
  • Zhu, H. T. and Zhang, H. P. (2006). Asymptotics for estimation and testing procedures under loss of identifiability. J. Multivariate Anal. 97 19–45.