## The Annals of Statistics

### Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes

#### Abstract

In this paper, we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models, with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model. We give two major applications: the case where the number of populations has to be tested in a mixture and the case of stationary ARMA$(p, q)$ processes where the order $(p, q)$ has to be tested. We give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model. In the case of order selection for ARMAs, the asymptotic distribution is invariant with respect to the parameters generating the process. A locally conic parametrization is a key tool in deriving the limiting distributions; it allows one to discover the deep similarity between the two problems.

#### Article information

Source
Ann. Statist. Volume 27, Number 4 (1999), 1178-1209.

Dates
First available in Project Euclid: 4 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1017938921

Digital Object Identifier
doi:10.1214/aos/1017938921

Mathematical Reviews number (MathSciNet)
MR1740115

Zentralblatt MATH identifier
0957.62073

#### Citation

Dacunha-Castelle, D.; Gassiat, E. Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes. Ann. Statist. 27 (1999), no. 4, 1178--1209. doi:10.1214/aos/1017938921. http://projecteuclid.org/euclid.aos/1017938921.

#### References

• AZENCOTT, R. and DACUNHA-CASTELLE, D. 1986. Series of Irregular Observations Forecasting and Model Building. Springer, Berlin. Z.
• BERAN, R. and MILLAR, P. W. 1987. Stochastic estimation and testing. Ann. Statist. 15 1131 1154. Z.
• BICKEL, P. and CHERNOFF, H. 1993. Asymptotic distribution of the likelihood ratio statistic in a prototypical non regular problem. In Statistics and Probability: A Raghu Raj BaZ hadur Festschrift J. K. Ghosh, S. K. Mitra, K. R. Parthasarathy and B. L. S. Prakasa. Rao, eds. 83 96. Wiley, New York. Z.
• 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.
• CIUPERCA, G. 1998. Maximum likelihood test for mixtures on translation parameter. Preprint, Universite d'Orsay. ´ Z. DACUNHA-CASTELLE, D. and GASSIAT, E. 1997. Testing in locally conic models and application Z. to mixture models. ESAIM: P & S Probability and Statistics 285 317. http: www. edpsciencs.com ps. Z.
• DAHLHAUS, R. 1988. Empirical processes and their applications to time series analysis. Stochastic Process. Appl. 30 69 83. Z.
• FEDER, P. I. 1975. The log likelihood ratio in segmented regression. Ann. Statist. 3 84 97. Z.
• GHOSH, J. K. and SEN, P. K. 1985. On the asymptotic performance of the log-likelihood ratio statistic for the mixture model and related results. Proc. Berkeley Conf. in Honor of Z. Jerzy Neyman and Jack Kiefer L. M. Le Cam and R. A. Olshen, eds. 2 789 806. Wadsworth, Belmont, CA. Z.
• HANNAN, J. 1980. The estimation of the order of an Arma process. Ann. Statist. 8 1071 1081. Z.
• HARTIGAN, J. A. 1985. A failure of likelihood asymptotics for normal mixtures. In Proc. Berkeley Z Conf. in Honor of Jerzy Neyman and Jack Kiefer L. M. Le Cam and R. A. Olshen,. eds. 2 807 810. Wadsworth, Belmont, CA. Z.
• KERIBIN, C. 1997. Consistent estimation of the order of mixture models. Preprint 61, Evry. ´ Z.
• LEMDANI, M. and PONS, O. 1997. Likelihood ratio tests for genetic linkage. Statist. Probab. Lett. 33 15 22. Z.
• LINDSAY, B. G. 1995. Mixture Models: Theory, Geometry and Applications. IMS, Hayward, CA. Z.
• POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, Berlin. Z.
• TEICHER, H. 1965. Identifiability of finite mixtures. Ann. Math. Statist. 36 423 439. Z.
• VAN DER VART, A. W. and WELLNER, J. A. 1996. Empirical Processes. Springer, Berlin. Z.
• VERES, S. 1987. Asymptotic distribution of likelihood ratios for overparametrized Arma processes. J. Time Ser. Anal. 8 345 357. Z.
• YAKOWITZ, S. J. and SPRAGINS, J. D. 1968. On the identifiability of finite mixtures. Ann. Math. Statist. 39 209 214.
• PROBABILITES, STATISTIQUE ET MODELISATION ´ ´ UNIVERSITE DE PARIS-SUD ´ BATIMENT 425 91405 ORSAY CEDEX FRANCE