Adaptive confidence bands for Markov chains and diffusions: Estimating the invariant measure and the drift

Söhl, Jakob; Trabs, Mathias

ESAIM: Probability and Statistics, Tome 20 (2016), pp. 432-462.

As a starting point we prove a functional central limit theorem for estimators of the invariant measure of a geometrically ergodic Harris-recurrent Markov chain in a multi-scale space. This allows to construct confidence bands for the invariant density with optimal (up to undersmoothing) $L^{\infty }$-diameter by using wavelet projection estimators. In addition our setting applies to the drift estimation of diffusions observed discretely with fixed observation distance. We prove a functional central limit theorem for estimators of the drift function and finally construct adaptive confidence bands for the drift by using a completely data-driven estimator.

R. Adamczak and W. Bednorz, Exponential concentration inequalities for additive functionals of Markov chains. ESAIM: PS 19 (2015) 440–481. | DOI | Numdam | Zbl

Y. Aït-Sahalia, Econometrics of Diffusion Models. John Wiley & Sons, Ltd. (2010).

D. Bakry, P. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254 (2008) 727–759. | DOI | Zbl

R.F. Bass, Diffusions and elliptic operators. Springer, New York (1998). | Zbl

A.D. Bull, Honest adaptive confidence bands and self-similar functions. Electron. J. Stat. 6 (2012) 1490–1516. | Zbl

J.V. Castellana and M.R. Leadbetter, On smoothed probability density estimation for stationary processes. Stochastic Process. Appl. 21 (1986) 179–193. | DOI | Zbl

I. Castillo and R. Nickl, Nonparametric Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist. 41 (2013) 1999–2028. | DOI | Zbl

I. Castillo and R. Nickl, On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures. Ann. Statist. 42 (2014) 1941–1969. | DOI | Zbl

D. Chauveau and J. Diebolt, Estimation of the asymptotic variance in the CLT for Markov chains. Stoch. Models 19 (2003) 449–465. | DOI | Zbl

X. Chen, Limit theorems for functionals of ergodic Markov chains with general state space. Mem. Amer. Math. Soc. 139 (1999) 664. | Zbl

V. Chernozhukov, D. Chetverikov and K. Kato, Anti-concentration and honest, adaptive confidence bands. Ann. Statist. 42 (2014) 1787–1818. | DOI | Zbl

J. Chorowski and M. Trabs, Spectral estimation for diffusions with random sampling times. Stochastic Process. Appl. 126 (2016) 2976–3008. | DOI | Zbl

F. Comte, V. Genon-Catalot and Y. Rozenholc, Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli 13 (2007) 514–543. | DOI | Zbl

A. Dalalyan, Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 (2005) 2507–2528. | DOI | Zbl

L. Galtchouk and S. Pergamenshchikov, Geometric ergodicity for classes of homogeneous Markov chains. Stochastic Process. Appl. 124 (2014) 3362–3391. | DOI | Zbl

C.J. Geyer, Practical Markov chain Monte Carlo. Stat. Sci. 7 (1992) 473–483.

I.I. Gihman and A.V. Skorohod, Stochastic differential equations. Springer, Heidelberg (1972). | Zbl

E. Giné and R. Nickl, Confidence bands in density estimation. Ann. Statist. 38 (2010) 1122–1170. | DOI | Zbl

E. Giné and R. Nickl, Mathematical foundations of infinite-dimensional statistical models. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press (2015).

E. Gobet, M. Hoffmann and M. Reiß, Nonparametric estimation of scalar diffusions based on low frequency data. Ann. Statist. 32 (2004) 2223–2253. | DOI | Zbl

S. Gugushvili and P. Spreij, Nonparametric Bayesian drift estimation for multidimensional stochastic differential equations. Lith. Math. J. 54 (2014) 127–141. | DOI | Zbl

L.P. Hansen and J.A. Scheinkman, Back to the future: generating moment implications for continuous-time Markov processes. Econometrica 63 (1995) 767–804. | DOI | Zbl

L.P. Hansen, J.A. Scheinkman and N. Touzi, Spectral methods for identifying scalar diffusions. J. Econom. 86 (1998) 1–32. | DOI | Zbl

W. Härdle, G. Kerkyacharian, D. Picard and A.Tsybakov, Wavelets, approximation, and statistical applications. Vol. 129 of Lecture Notes in Statistics. Springer-Verlag, New York (1998). | Zbl

M. Hoffmann, Adaptive estimation in diffusion processes. Stochastic Process. Appl. 79 (1999) 135–163. | DOI | Zbl

M. Hoffmann and R. Nickl, On adaptive inference and confidence bands. Ann. Statist. 39 (2011) 2383–2409. | DOI | Zbl

D. Kristensen, Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. J. Econom. 156 (2010) 239–259. | DOI | Zbl

Y.A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer Series in Statistics. Springer-Verlag London, Ltd., London (2004). | Zbl

C. Lacour, Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stochastic Process. Appl. 118 (2008) 232–260. | DOI | Zbl

M. Ledoux, The concentration of measure phenomenon. Vol. 89 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2001). | Zbl

E. Löcherbach, D. Loukianova and O. Loukianov, Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process. ESAIM: PS 15 (2011) 197–216. | DOI | Zbl

M. Glow, On nonparametric confidence intervals. Ann. Statist. 25 (1997) 2547–2554. | Zbl

H. Masuda, I. Negri and Y. Nishiyama, Goodness-of-fit test for ergodic diffusions by discrete-time observations: an innovation martingale approach. J. Nonparametr. Stat. 23 (2011) 237–254. | DOI | Zbl

S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability. Cambridge University Press, Cambridge (2009). | Zbl

I. Negri and Y. Nishiyama, Goodness of fit test for ergodic diffusions by tick time sample scheme. Stat. Inference Stoch. Process. 13 (2010) 81–95. | DOI | Zbl

R. Nickl and J. Söhl, Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions. To appear in Ann. Statist. (2016).

C.P. Robert, Convergence control methods for Markov chain Monte Carlo algorithms. Statistical Science 10 (1995) 231–253. | DOI | Zbl

M. Rosenblatt, Density estimates and Markov sequences. In Nonparametric Techniques in Statistical Inference (Proc. Sympos., Indiana Univ., Bloomington, Ind., 1969). Cambridge Univ. Press, London (1970) 199–213.

G.G. Roussas, Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 (1969) 73–87. | DOI | Zbl

E. Schmisser, Nonparametric estimation of the derivatives of the stationary density for stationary processes. ESAIM: PS 17 (2013) 33–69. | DOI | Numdam | Zbl

V.G. Spokoiny, Adaptive drift estimation for nonparametric diffusion model. Ann. Statist. 28 (2000) 815–836. | DOI | Zbl

B. Szabó, A.W. Van Der Vaart and J.H. Van Zanten, Frequentist coverage of adaptive nonparametric Bayesian credible sets (with discussion). Ann. Statist. 43 (2015) 1391–1428. | Zbl

H. Triebel, Theory of function spaces. Vol. 78 of Monographs in Mathematics. Birkhäuser Verlag, Basel (1983). | Zbl

F. Van Der Meulen and H. Van Zanten, Consistent nonparametric Bayesian inference for discretely observed scalar diffusions. Bernoulli 19 (2013) 44–63. | DOI | Zbl

A. Van Der Vaart and H. Van Zanten, Donsker theorems for diffusions: necessary and sufficient conditions. Ann. Probab. 33 (2005) 1422–1451. | DOI | Zbl

A. W. van der Vaart, Asymptotic statistics. Vol. 3 of Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (1998). | Zbl

S. Yakowitz, Nonparametric density and regression estimation for Markov sequences without mixing assumptions. J. Multivariate Anal. 30 (1989) 124–136. | DOI | Zbl