We prove the global asymptotic equivalence between the experiments generated by the discrete (high frequency) or continuous observation of a path of a time inhomogeneous jump-diffusion process and a Gaussian white noise experiment. Here, the parameter of interest is the drift function and the observation time can be both bounded or diverging. The approximation is given in the sense of the Le Cam -distance, under some smoothness conditions on the unknown drift function. These asymptotic equivalences are established by constructing explicit Markov kernels that can be used to reproduce one experiment from the other.
DOI : 10.1051/ps/2015005
Mots clés : Non-parametric experiments, Le Cam distance, asymptotic equivalence, Lévy processes, additive processes, white noise
@article{PS_2015__19__560_0,
author = {Mariucci, Ester},
title = {Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise},
journal = {ESAIM: Probability and Statistics},
pages = {560--577},
publisher = {EDP-Sciences},
volume = {19},
year = {2015},
doi = {10.1051/ps/2015005},
mrnumber = {3433426},
zbl = {1331.62058},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2015005/}
}
TY - JOUR AU - Mariucci, Ester TI - Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise JO - ESAIM: Probability and Statistics PY - 2015 SP - 560 EP - 577 VL - 19 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2015005/ DO - 10.1051/ps/2015005 LA - en ID - PS_2015__19__560_0 ER -
%0 Journal Article %A Mariucci, Ester %T Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise %J ESAIM: Probability and Statistics %D 2015 %P 560-577 %V 19 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2015005/ %R 10.1051/ps/2015005 %G en %F PS_2015__19__560_0
Mariucci, Ester. Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 560-577. doi : 10.1051/ps/2015005. http://www.numdam.org/articles/10.1051/ps/2015005/
and , Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 (1996) 2384–2398. | MR | Zbl
and , Asymptotic nonequivalence of nonparametric experiments when the smoothness index is . Ann. Statist. 26 (1998) 279–287. | MR | Zbl
, , and , Asymptotic equivalence theory for nonparametric regression with random design. Dedicated to the memory of Lucien Le Cam. Ann. Statist. 30 (2002) 688–707. | MR | Zbl
, , and , Equivalence theory for density estimation, Poisson processes and Gaussian white noise with drift. Ann. Statist. 32 (2004) 2074–2097. | MR | Zbl
and , Limit experiments of GARCH. Bernoulli 18 (2012) 64–99. | MR | Zbl
, Deficiency distance between multinomial and multivariate normal experiments. Dedicated to the memory of Lucien Le Cam. Ann. Statist. 30 (2002) 708–730. | MR | Zbl
, A continuous Gaussian approximation to a nonparametric regression in two dimensions. Bernoulli 12 (2006) 143–156. | MR | Zbl
, Asymptotic approximation of nonparametric regression experiments with unknown variances. Ann. Statist. 35 (2007) 1644–1673. | MR | Zbl
, Asymptotically sufficient statistics in nonparametric regression experiments with correlated noise. J. Probab. Stat. 19 (2009) 275308. | MR | Zbl
R. Cont and P. Tankov, Financial modelling with jump processes. Chapman & Hall/CRC Financial Mathematics Series. Chapman & Hall/CRC, Boca Raton, FL (2004). | MR | Zbl
and , Asymptotic statistical equivalence for scalar ergodic diffusions. Probab. Theory Relat. Fields 134 (2006) 248–282. | MR | Zbl
and , Asymptotic statistical equivalence for ergodic diffusions: the multidimensional case. Probab. Theory Relat. Fields 137 (2007) 25–47. | MR | Zbl
and , Asymptotic equivalence for a null recurrent diffusion. Bernoulli 8 (2002) 139–174. | MR | Zbl
et al. Statistical inference across time scales. Electron. J. Stat. 5 (2011) 2004–2030. | MR | Zbl
and , Asymptotic equivalence of nonparametric regression and white noise model has its limits. Stat. Probab. Lett. 28 (1996) 143–145. | MR | Zbl
P. Etoré, S. Louhichi and E. Mariucci, Asymptotic equivalence of jumps lévy processes and their discrete counterpart. Preprint arXiv:1305.6725 (2013).
and , Asymptotic equivalence of nonparametric diffusion and Euler scheme experiments. Ann. Stat. 42 (2014) 1145–1165. | MR | Zbl
, and , Asymptotic equivalence of estimating a Poisson intensity and a positive diffusion drift. Ann. Stat. 30 (2002) 731–753. Dedicated to the memory of Lucien Le Cam. | MR | Zbl
, and , Asymptotic equivalence of spectral density estimation and Gaussian white noise. Ann. Stat. 38 (2010) 181–214. | MR | Zbl
and , Asymptotic equivalence for nonparametric generalized linear models. Probab. Theory Related Fields 111 (1998) 167–214. | MR | Zbl
and , Asymptotic equivalence for nonparametric regression. Math. Methods Stat. 11 (2002) 1–36. | MR | Zbl
and , Asymptotic equivalence of nonparametric autoregression and nonparametric regression. Ann. Stat. 34 1701–1732 (2006). | MR | Zbl
and , Asymptotic equivalence for a model of independent non identically distributed observations. Statist. Decisions 21 (2003) 197–218. | MR | Zbl
L. Le Cam, Théorie asymptotique de la décision statistique. Séminaire de Mathématiques Supérieures, no. 33 (Été, 1968). Les Presses de l’Université de Montréal, Montreal, Québec (1969). | MR | Zbl
Lucien Le Cam, Asymptotic methods in statistical decision theory. Springer Ser. Stat. Springer-Verlag, New York (1986). | MR | Zbl
L. Le Cam and G. Lo Yang, Asymptotics in statistics. Some basic concepts. Springer Ser. Stat. Springer-Verlag, New York, 2nd edn. (2000). | MR
E. Mariucci, Asymptotic equivalence of discretely observed diffusion processes and their euler scheme: small variance case. To appear in Stat. Inference Stoch. Process (2015). DOI:. | DOI | MR
, Asymptotic equivalence of functional linear regression and a white noise inverse problem. Ann. Stat. 39 (2011) 1471–1495. | MR | Zbl
and , Asymptotic equivalence for nonparametric regression with non-regular errors. Probab. Theory Related Fields 155 (2013) 201–229. | MR | Zbl
and , Diffusion approximation for nonparametric autoregression. Probab. Theory Related Fields 112 (1998) 535–543. | MR | Zbl
, Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Stat. 24 (1996) 2399–2430. | MR | Zbl
, Asymptotic equivalence for nonparametric regression with multivariate and random design. Ann. Stat. 36 (2008) 1957–1982. | MR | Zbl
, Asymptotic equivalence for inference on the volatility from noisy observations. Ann. Stat. 39 (2011) 772–802. | MR | Zbl
, On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. Statist. Decisions 2 (2004) 235–243. | MR | Zbl
H. Strasser, Mathematical theory of statistics. Statistical experiments and asymptotic decision theory. Vol. 7 of de Gruyter Stud. Math. Walter de Gruyter & Co., Berlin (1985). | MR | Zbl
A.B. Tsybakov, Introduction to nonparametric estimation. Springer Ser. Stat. Springer, New York (2009). Revised and extended from the 2004 French original, Translated by Vladimir Zaiats. | MR
, Asymptotic nonequivalence of Garch models and diffusions. Ann. Stat. 30 (2002) 754–783. Dedicated to the memory of Lucien Le Cam. | MR | Zbl
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