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Testing in locally conic models, and application to mixture models. (English) Zbl 1007.62507

Summary: We address the problem of testing hypotheses using maximum likelihood statistics in non-identifiable models. We derive the asymptotic distribution under very general assumptions. The key idea is a local reparameterization, depending on the underlying distribution, which is called locally conic. This method enlights how the general model induces the structure of the limiting distribution in terms of dimensionality of some derivative space. We present various applications of the theory. The main application is to mixture models. Under very general assumptions, we solve completely the problem of testing the size of the mixture using maximum likelihood statistics. We derive the asymptotic distribution of the maximum likelihood statistic ratio which takes an unexpected form.

MSC:

62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
62G10 Nonparametric hypothesis testing
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[1] ADLER, R. J. ( 1990). An introduction to continuity, extrema, and related topics for general Gaussian processes. IMS Lecture Notes-Monograph Series. Zbl0747.60039 MR1088478 · Zbl 0747.60039
[2] AZAIS, J. M. and WSCHEBOR, M. ( 1995). A formula to compute the distribution of the maximum of a random process. Université P. Sabatier, Toulouse.
[3] AZENCOTT, R. and DACUNHA-CASTELLE, D. ( 1984). Séries d’observations irrégulières. Masson. Zbl0546.62060 MR746133 · Zbl 0546.62060
[4] BERAN, R. and MILLAR, P. W. ( 1987). Stochastic estimation and testing. Annals of Stat., 15 1131-1154. Zbl0644.62028 MR902250 · Zbl 0644.62028
[5] BERDAI, A. and GAREL, B. ( 1994). Performances d’un test d’homogénéité contre une hypothèse de mélange gaussien. Rev. Stat. Appl. 42 63-79. Zbl0972.62505 MR1278467 · Zbl 0972.62505
[6] 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 Bahadur Festschrift.
[7] DACUNHA-CASTELLE, D. and DUFLO, M. ( 1986). Probability and Statistics. Springer Verlag New-York. Zbl0586.62003 · Zbl 0586.62003
[8] DACUNHA-CASTELLE, D. and GASSIAT, E. ( 1996). Testing the order of a model using locally conic parametrization: population mixtures and stationary ARMA processes. Submitted. Zbl0957.62073 · Zbl 0957.62073
[9] DONOHO, D. L. ( 1988). One-sided inference about functionals of a density. Annals of Stat. 16 1390-1420. Zbl0665.62040 MR964930 · Zbl 0665.62040
[10] DUDLEY, R. M. ( 1967). The size of compact subsets of hilbert space and continuity of Gaussian processes. Journal of Funct. Analysis 1 290-330. Zbl0188.20502 MR220340 · Zbl 0188.20502
[11] GHOSH, J. and SEN, P. ( 1985). On the asymptotic performance of the log-likelihood ratio statistic for the mixture model and related results. In Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer. Le Cam, L.M. and Olshen, R.A. eds. MR822065 · Zbl 1373.62075
[12] HANNAN, E. J. ( 1982). Testing for autocorrelation and akaike’s criterion. In Essays in Statistical Science, p. 403-412. Gani, J.M., Hannan, E.J. eds. Zbl0498.62078 MR633209 · Zbl 0498.62078
[13] HARTIGAN, J. A. ( 1985). A failure of likelihood ratio asymptotics for normal mixtures. In Proceedings of the Berkeley conference in honor of Jerzy Neyman and Jack Kiefer. Le Cam, L.M. and Olshen, R.A. eds. · Zbl 1373.62070
[14] LAURENT, B. ( 1993). Estimation de fonctionnelles intégrales non linéaires de la densité et de ses dérivées. PhD thesis, Université de Paris XI, France.
[15] OSSIANDER, M. ( 1987). A central limit theorem under metric entropy with l2 bracketing. Annals of Prob. 15 897-919. Zbl0665.60036 MR893905 · Zbl 0665.60036
[16] REDNER, R. ( 1981). Note on the consistency of the maximum likelihood estimate for nonidentifiable distributions. Annals of Stat. 9 225-228. Zbl0453.62021 MR600553 · Zbl 0453.62021
[17] ROUSSAS, G. G. ( 1970). Contiguity of probability measures: some applications in statistics. Princeton University press. Zbl1152.60007 MR359099 · Zbl 1152.60007
[18] SELF, S. and LIANG, K. ( 1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Jour. Amer. Stat. Assoc. 823 605-610. Zbl0639.62020 MR898365 · Zbl 0639.62020
[19] TEICHER, H. ( 1965). Identifiability of finite mixtures. Annals of Math. Statist. 36 423-439. MR155376 · Zbl 0219.60031
[20] VAN DER VAART, A. W. and WELLNER, J. A. ( 1996). Empirical Processes. Springer Verlag. · Zbl 0862.60002
[21] YAKOWITZ, S. J. and SPRAGINS, J. D. ( 1968). On the identifiability of finite mixtures. Annals of Math. Stat. 39 209-214. Zbl0155.25703 MR224204 · Zbl 0155.25703
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