Parametric first-order Edgeworth expansion for Markov additive functionals. Application to $M$-estimations

Ferré, D.

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015), p. 781-808 / Harvested from Numdam

Résumé

Grâce à une approche spectrale, nous donnons des conditions assurant la validité du développement d’Edgeworth d’ordre 1 paramétrique, dans le cadre général des fonctionnelles bivariées et additives de chaînes de Markov fortement ergodiques. En particulier, soit $(X_{n})_{n\in \mathbb {N} }$ une chaîne de Markov $V$-géométriquement ergodique dont la loi dépend d’un paramètre $\theta $. Nous montrons alors que $\lbrace \xi _{p}(X_{n-1},X_{n});p\in \mathcal {P} ,n\ge 1\rbrace $ satisfait un développement d’Edgeworth d’ordre 1 uniforme (en $(\theta ,p)$) si $\lbrace \xi _{p}(\cdot ,\cdot );p\in \mathcal {P} \rbrace $ satisfait une condition de type non-lattice ainsi qu’une condition quasi-optimale de moment-domination. De plus, ce résultat est établi dans le cas où les données $(X_{n})_{n\in \mathbb {N} }$ ne sont pas nécessairement stationnaires. Ce résultat est appliqué en particulier aux $M$-estimateurs associés à des chaînes de Markov $V$-géométriquement ergodiques. Les $M$-estimateurs de processus autorégressifs sont étudiés.

Abstract

We give a spectral approach to prove a parametric first-order Edgeworth expansion for bivariate additive functionals of strongly ergodic Markov chains. In particular, given any $V$-geometrically ergodic Markov chain $(X_{n})_{n\in \mathbb {N} }$ whose distribution depends on a parameter $\theta $, we prove that $\lbrace \xi _{p}(X_{n-1},X_{n});p\in \mathcal {P} ,n\ge 1\rbrace $ satisfies a uniform (in $(\theta ,p)$) first-order Edgeworth expansion provided that $\lbrace \xi _{p}(\cdot ,\cdot );p\in \mathcal {P} \rbrace $ satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence $(X_{n})_{n\in \mathbb {N} }$ need not be stationary. This result is applied to $M$-estimators of Markov chains and in particular of $V$-geometrically ergodic Markov chains. The $M$-estimators of some autoregressive processes are studied.

Détail

Publié le : 2015-01-01

MR 3335025

DOI : https://doi.org/10.1214/13-AIHP592
Classification:  60F05,  60J05,  62F12,  62M05

Bibliographie

[1] V. Baladi. Positive Transfer Operators and Decay of Correlations. Adv. Ser. Nonlinear Dynam. 16. World Scientific Publishing Co., River Edge, NJ, 2000. | MR 1793194 | Zbl 1012.37015

[2] P. Bertail and S. Clémençon. A renewal approach to markovian $U$-statistics. Math. Methods Statist. 20 (2011) 79–105. | MR 2882153 | Zbl 06414573

[3] P. Billingsley. Statistical Inference for Markov Processes. Univ. Chicago Press, Chicago, IL, 1961. | MR 123419 | Zbl 0106.34201

[4] E. Bolthausen. The Berry–Esseen theorem for strongly mixing Harris recurrent Markov chains. Probab. Theory Related Fields 60 (1982) 283–289. | MR 664418 | Zbl 0476.60022

[5] R. Dahlhaus. Efficient parameter estimation for self similar processes. Ann. Statist. 17 (1989) 1749–1766. | MR 1026311 | Zbl 0703.62091

[6] D. Dehay and J. F. Yao. On likelihood estimation for discretely observed Markov jump processes. Aust. N. Z. J. Stat. 49 (2007) 93–107. | MR 2345413 | Zbl 1117.62082

[7] J. Durbin. Approximations for densities of sufficient estimators. Biometrika 67 (1980) 311–333. | MR 581728 | Zbl 0436.62020

[8] W. Feller. An Introduction to Probability Theory and Its Applications. Vol. II. Wiley, New York, 1971. | MR 270403 | Zbl 0219.60003

[9] D. Ferré. Développements d’Edgeworth en statistique des modèles markoviens. Ph.D. thesis, Institut National des Sciences Appliquées de Rennes.

[10] D. Ferré. Développement d’Edgeworth d’ordre 1 pour des $M$-estimateurs dans le cas de chaînes $V$-géométriquement ergodiques. C. R. Math. Acad. Sci. Paris 348 (2010) 331–334. | MR 2600134 | Zbl 1186.62103

[11] D. Ferré and D. Guibourg. Regularity of the characteristic function of additive functionals for iterated function systems. Statistical applications. Markov Process. Related Fields 19 (2013) 299–342. | MR 3113946 | Zbl 1306.60109 | Zbl 06238349

[12] D. Ferré, L. Hervé and J. Ledoux. Limit theorems for stationary Markov processes with $L^{2}$-spectral gap. Ann. Inst. Henri Poincaré Probab. Stat. 48 (2012) 396–423. | Numdam | MR 2954261 | Zbl 1245.60068

[13] C.-D. Fuh. Efficient likelihood estimation in state space models. Ann. Statist. 34 (2006) 2026–2068. | MR 2283726 | Zbl 1246.62185

[14] P. Gänssler. Note on minimum contrast estimators for Markov processes. Metrika 19 (1972) 115–130. | MR 418374 | Zbl 0251.62057

[15] F. Götze and C. Hipp. Asymptotic distribution of statistics in time series. Ann. Statist. 22 (1994) 2062–2088. | MR 1329183 | Zbl 0827.62015

[16] H. Hennion and L. Hervé. Limit Theorems for Markov Chains and Stochastic Properties of Dynamical Systems by Quasi-Compactness. Lecture Notes Math. 1766. Springer, Berlin, 2001. | MR 1862393 | Zbl 0983.60005

[17] L. Hervé, J. Ledoux and V. Patilea. A Berry-Esseen theorem on $M$-estimators for $V$-geometrically ergodic Markov chains. Bernoulli 18 (2012) 703–734. | MR 2922467 | Zbl 1279.60089

[18] L. Hervé and F. Pène. The Nagaev–Guivarc’h method via the Keller–Liverani theorem. Bull. Soc. Math. France 138 (3) (2010) 415–489. | Numdam | MR 2729019 | Zbl 1205.60133

[19] J. L. Jensen. Asymptotic expansions for strongly mixing Harris recurrent Markov chains. Scand. J. Statist. 16 (1989) 47–63. | MR 1003968 | Zbl 0674.60067

[20] G. Keller and C. Liverani. Stability of the spectrum for transfer operators. Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) XXVIII (1999) 141–152. | Numdam | MR 1679080 | Zbl 0956.37003

[21] O. Lieberman, J. Rousseau and D. M. Zucker. Valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary, Gaussian, strongly dependent process. Ann. Statist. 31 (2003) 586–612. | MR 1983543 | Zbl 1067.62021

[22] C. Liverani. Invariant measures and their properties. A functional analytic point of view. In Dynamical Systems. Part II 185–237. Cent. Ric. Mat. Ennio Giorgi. Scuola Norm. Sup., Pisa, 2003. | MR 2071241 | Zbl 1066.37013

[23] V. K. Malinovskii. Limit theorems for Harris Markov chains. I. Theory Probab. Appl. 31 (1987) 269–285. | MR 850991 | Zbl 0657.60087

[24] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, Berlin, 1993. | MR 1287609 | Zbl 0925.60001

[25] J. Pfanzagl. Asymptotic expansions related to minimum contrast estimators. Ann. Statist. 1 (6) (1973) 993–1026. | MR 359151 | Zbl 0273.62015

[26] B. L. S. Prakasa Rao. Maximum likelihood estimation for Markov processes. Ann. Inst. Statist. Math. 24 (1972) 333–345. | MR 336936 | Zbl 0327.62054

[27] F. Räbiger and M. P. H. Wolff. On the approximation of positive operators and the behaviour of the spectra of the approximants. Integral Equations Operator Theory 28 (1) (1997) 72–86. | MR 1446832 | Zbl 0901.47009

[28] G. G. Roussas. Asymptotic inferencer in Markov process. Ann. Math. Statist. 36 (1965) 978–982. | MR 179882 | Zbl 0138.11601

[29] W. L. Smith. Regenerative stochastic processes. Proc. Roy. Soc. London. Ser. A. 232 (1955) 6–31. | MR 73877 | Zbl 0067.36301