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) no. 2, pp. 781-808.

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 ${\left(X_n\right)}_{n\in \mathbb{N}}$ une chaîne de Markov $V$-géométriquement ergodique dont la loi dépend d’un paramètre $\theta$. Nous montrons alors que $\{{\xi }_p(X_{n-1},X_n);p\in 𝒫,n\ge 1\}$ satisfait un développement d’Edgeworth d’ordre 1 uniforme (en $(\theta ,p)$) si $\{{\xi }_p(·,·);p\in 𝒫\}$ 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 ${\left(X_n\right)}_{n\in \mathbb{N}}$ 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.

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 ${\left(X_n\right)}_{n\in \mathbb{N}}$ whose distribution depends on a parameter $\theta$, we prove that $\{{\xi }_p(X_{n-1},X_n);p\in 𝒫,n\ge 1\}$ satisfies a uniform (in $(\theta ,p)$) first-order Edgeworth expansion provided that $\{{\xi }_p(·,·);p\in 𝒫\}$ satisfies some non-lattice condition and an almost optimal moment domination condition. Furthermore, the sequence ${\left(X_n\right)}_{n\in \mathbb{N}}$ 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.

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

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

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

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

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

[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. | DOI | MR | Zbl

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

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

[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. | DOI | MR | Zbl

[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 | Zbl

[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. | DOI | EuDML | Numdam | MR | Zbl

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

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

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

[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. | DOI | MR | Zbl

[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. | DOI | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

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

[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. | EuDML | Numdam | MR | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

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