## Testing the order of a model.(English)Zbl 1096.62016

Summary: This paper deals with order identification for nested models in the i.i.d. framework. We study the asymptotic efficiency of two generalized likelihood ratio tests of the order. They are based on two estimators which are proved to be strongly consistent. A version of Stein’s lemma yields an optimal underestimation error exponent. The lemma also implies that the overestimation error exponent is necessarily trivial. Our tests admit nontrivial underestimation error exponents. The optimal underestimation error exponent is achieved in some situations. The overestimation error can decay exponentially with respect to a positive power of the number of observations.
These results are proved under mild assumptions by relating the underestimation (resp. overestimation) error to large (resp. moderate) deviations of the log-likelihood process. In particular, it is not necessary that the classical Cramér condition be satisfied; namely, the log-densities are not required to admit every exponential moment. Three benchmark examples with specific difficulties (location mixture of normal distributions, abrupt changes and various regressions) are detailed so as to illustrate the generality of our results.

### MSC:

 62F05 Asymptotic properties of parametric tests 60F10 Large deviations 62C99 Statistical decision theory 60G57 Random measures 62F12 Asymptotic properties of parametric estimators 46N30 Applications of functional analysis in probability theory and statistics
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### References:

 [1] Akaike, H. (1974). A new look at the statistical model identification. IEEE Trans. Automatic Control 19 716–723. · Zbl 0314.62039 [2] Azencott, R. and Dacunha-Castelle, D. (1986). Series of Irregular Observations . Springer, New York. · Zbl 0593.62088 [3] Bahadur, R. R. (1967). An optimal property of the likelihood ratio statistic. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 13–26. Univ. California Press, Berkeley. · Zbl 0211.50901 [4] Bahadur, R. R. (1971). Some Limit Theorems in Statistics . SIAM, Philadelphia. · Zbl 0257.62015 [5] Bahadur, R. R., Zabell, S. L. and Gupta, J. C. (1980). Large deviations, tests, and estimates. In Asymptotic Theory of Statistical Tests and Estimation (I. M. Chakravarti, ed.) 33–64. Academic Press, New York. · Zbl 0601.62037 [6] Barron, A., Birgé, L. and Massart, P. (1999). Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 301–413. · Zbl 0946.62036 [7] Boucheron, S. and Gassiat, E. (2005). An information-theoretic perspective on order estimation. In Inference in Hidden Markov Models (O. Cappé, E. Moulines and T. Rydén, eds.) 565–601. Springer, New York. [8] Boucheron, S. and Gassiat, E. (2006). Error exponents for AR order testing. IEEE Trans. Inform. Theory 52 472–488. · Zbl 1284.62545 [9] Čencov, N. N. (1982). Statistical Decision Rules and Optimal Inference . Amer. Math. Soc., Providence, RI. [10] Chernoff, H. (1956). Large sample theory: Parametric case. Ann. Math. Statist. 27 1–22. · Zbl 0072.35703 [11] Csiszár, I. (1975). $$I$$-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 146–158. · Zbl 0318.60013 [12] Csiszár, I. (2002). Large-scale typicality of Markov sample paths and consistency of MDL order estimators. IEEE Trans. Inform. Theory 48 1616–1628. · Zbl 1060.62092 [13] Csiszár, I. and Körner, J. (1981). Information Theory : Coding Theorems for Discrete Memoryless Systems . Academic Press, New York. · Zbl 0568.94012 [14] Csiszár, I. and Shields, P. C. (2000). The consistency of the BIC Markov order estimator. Ann. Statist. 28 1601–1619. · Zbl 1105.62311 [15] Dacunha-Castelle, D. and Gassiat, E. (1997). The estimation of the order of a mixture model. Bernoulli 3 279–299. · Zbl 0889.62012 [16] 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. · Zbl 0957.62073 [17] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications , 2nd ed. Springer, New York. · Zbl 0896.60013 [18] Dudley, R. M. and Philipp, W. (1983). Invariance principles for sums of Banach space valued random elements and empirical processes. Z. Wahrsch. Verw. Gebiete 62 509–552. · Zbl 0488.60044 [19] Dupuis, P. and Ellis, R. S. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, New York. · Zbl 0904.60001 [20] Finesso, L., Liu, C.-C. and Narayan, P. (1996). The optimal error exponent for Markov order estimation. IEEE Trans. Inform. Theory 42 1488–1497. · Zbl 0856.62071 [21] Gassiat, E. (2002). Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. H. Poincaré Probab. Statist. 38 897–906. · Zbl 1011.62025 [22] Gassiat, E. and Boucheron, S. (2003). Optimal error exponents in hidden Markov models order estimation. IEEE Trans. Inform. Theory 49 964–980. · Zbl 1065.62148 [23] Guyon, X. and Yao, J. (1999). On the underfitting and overfitting sets of models chosen by order selection criteria. J. Multivariate Anal. 70 221–249. · Zbl 1070.62516 [24] Hannan, E. J., McDougall, A. J. and Poskitt, D. S. (1989). Recursive estimation of autoregressions. J. Roy. Statist. Soc. Ser. B 51 217–233. JSTOR: · Zbl 0675.62060 [25] Haughton, D. (1989). Size of the error in the choice of a model to fit data from an exponential family. Sankhyā Ser. A 51 45–58. · Zbl 0682.62021 [26] Hemerly, E. M. and Davis, M. H. A. (1991). Recursive order estimation of autoregressions without bounding the model set. J. Roy. Statist. Soc. Ser. B 53 201–210. JSTOR: · Zbl 0800.62538 [27] Henna, J. (1985). On estimating of the number of constituents of a finite mixture of continuous distributions. Ann. Inst. Statist. Math. 37 235–240. · Zbl 0577.62031 [28] Huber, P. J. (1967). The behavior of maximum likelihood estimates under nonstandard conditions. Proc. Fifth Berkeley Symp. Math. Statist. Probab. 1 221–233. Univ. California Press, Berkeley. · Zbl 0212.21504 [29] James, L. F., Priebe, C. E. and Marchette, D. J. (2001). Consistent estimation of mixture complexity. Ann. Statist. 29 1281–1296. · Zbl 1043.62023 [30] Keribin, C. (2000). Consistent estimation of the order of mixture models. Sankhyā Ser. A 62 49–66. · Zbl 1081.62516 [31] Keribin, C. and Haughton, D. (2003). Asymptotic probabilities of overestimating and underestimating the order of a model in general regular families. Comm. Statist. Theory Methods 32 1373–1404. · Zbl 1140.62312 [32] Léonard, C. and Najim, J. (2002). An extension of Sanov’s theorem. Application to the Gibbs conditioning principle. Bernoulli 8 721–743. · Zbl 1013.60018 [33] Leonardi, G. P. and Tamanini, I. (2002). Metric spaces of partitions, and Caccioppoli partitions. Adv. Math. Sci. Appl. 12 725–753. · Zbl 1044.49030 [34] Leroux, B. G. (1992). Consistent estimation of a mixing distribution. Ann. Statist. 20 1350–1360. · Zbl 0763.62015 [35] Mallows, C. L. (1973). Some comments on $$C_P$$. Technometrics 15 661–675. · Zbl 0269.62061 [36] Massart, P. (2000). Some applications of concentration inequalities to statistics. Probability theory. Ann. Fac. Sci. Toulouse Math. (6) 9 245–303. · Zbl 0986.62002 [37] Pollard, D. (1985). New ways to prove central limit theorems. Econometric Theory 1 295–314. [38] Rissanen, J. (1978). Modelling by shortest data description. Automatica 14 465–471. · Zbl 0418.93079 [39] Rockafellar, R. T. (1970). Convex Analysis . Princeton Univ. Press. · Zbl 0193.18401 [40] Schied, A. (1998). Cramer’s condition and Sanov’s theorem. Statist. Probab. Lett. 39 55–60. · Zbl 0911.60019 [41] Schwarz, G. (1978). Estimating the dimension of a model. Ann. Statist. 6 461–464. · Zbl 0379.62005 [42] Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985). Statistical Analysis of Finite Mixture Distributions . Wiley, Chichester. · Zbl 0646.62013 [43] van der Vaart, A. W. (1998). Asymptotic Statistics . Cambridge Univ. Press. · Zbl 0910.62001 [44] Wu, L. (1994). Large deviations, moderate deviations and LIL for empirical processes. Ann. Probab. 22 17–27. · Zbl 0793.60032
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