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Original Articles

An oracle inequality for penalised projection estimation of Lévy densities from high-frequency observations

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Pages 967-989 | Received 08 Apr 2010, Accepted 11 Apr 2011, Published online: 30 Jun 2011

References

  • Aït-Sahalia , Y. and Jacod , J. 2007 . Volatility Estimators for Discretely Sampled Lévy Processes . The Annals of Statistics , 35 : 355 – 392 .
  • Aït-Sahalia , Y. and Jacod , J. 2009 . Testing for Jumps in a Discretely Observed Process . The Annals of Statistics , 37 : 184 – 222 .
  • Akritas , M. G. 1982 . Asymptotic Theory for Estimating the Parameters of a Lévy Process . Annals of the Institute of Statistical Mathematics , 34 : 259 – 280 .
  • Barron , A. R. , Birgé , L. and Massart , P. 1999 . Risk Bounds for Model Selection via Penalization . Probability Theory and Related Fields , 113 : 301 – 413 .
  • Basawa , I. V. and Brockwell , P. J. 1982 . Non-Parametric Estimation for Non-decreasing Lévy processes . Journal of the Royal Statistical Society. Series B , 44 : 262 – 269 .
  • Belomestny , D. and Reiß , M. 2006 . Spectral Calibration of Exponential Lévy Models . Finance and Stochastics , 10 : 449 – 474 . doi:10.1007/s00780-006-0021-5
  • Billingsley , P. 1995 . Probability and Measure , 3 , New York : Wiley .
  • Birgé , L. and Massart , P. 1993 . Rates of Convergence for Minimum Contrast Estimators . Probability Theory and Related Fields , 97 : 113 – 150 .
  • Birgé , L. and Massart , P. 1997 . “ From Model Selection to Adaptive Estimation ” . In Festschrift for Lucien LeCam: Research Papers in Probability and Statistics , Edited by: Pollard , D. , Torgersen , E. and Yang , G. 55 – 87 . New York : Springer .
  • Birgé , L. and Massart , P. 1998 . Minimum Contrast Estimators on Sieves: Exponential Bounds and Rates of Convergence . Bernoulli , 4 : 329 – 375 .
  • Brenner , S. C. and Scott , L. R. 1994 . The Mathematical Theory of Finite Element Methods , New York : Springer .
  • Čencov , N. N. 1962 . Evaluation of an Unknown Distribution Density from Observations . Soviet Mathematics – Doklady , 3 : 1559 – 1562 .
  • Chen , S. X. , Delaigle , A. and Hall , P. 2010 . Nonparametric Estimation for a Class of Lévy Processes . Journal of Econometrics , 157 : 257 – 271 .
  • Comte , F. and Genon-Catalot , V. 2009 . Nonparametric Estimation for Pure Jump Lévy Processes Based on High Frequency Data . Stochastic Processes and their Applications , 119 : 4088 – 4123 .
  • Comte , F. and Genon-Catalot , V. 2010 . Nonparametric Adaptive Estimation for Pure Jump Lévy Processes . Annales de l'Institut Henri Poincaré. Probabilités et Statistiques , 46 : 595 – 617 .
  • Comte , F. and Genon-Catalot , V. 2011 . Estimation for Lévy Processes from High Frequency Data within a Long Time Intervals . The Annals of Statistics , 39 : 803 – 837 .
  • DeVore , R. A. and Lorentz , G. G. 1993 . Constructive Approximation , Berlin : Springer .
  • Figueroa-López , J. E. 2009 . “ Non-Parametric Estimation for Lévy Models Based on Discrete Sampling ” . In Optimality: The Third Erich L. Lehmann Symposium , Edited by: Rojo , J. 117 – 146 . Beachwood, OH : Institute of Mathematical Statistics . Vol. 57 of IMS Lecture Notes – Monograph Series
  • Figueroa-López , J. E. 2011 . Sieve-Based Confidence Intervals and Bands for Lévy Densities . Bernoulli , 17 : 643 – 670 .
  • Figueroa-López , J. E. and Houdré , C. 2006 . “ Risk Bounds for the Non-Parametric Estimation of Lévy Processes ” . In High Dimensional Probability: Proceedings of the Fourth International Conference , Edited by: Giné , E. , Kolchinskii , V. , Li , W. and Zinn , J. 96 – 116 . Beachwood, OH : Institute of Mathematical Statistics . Vol. 51 of IMS Lecture Notes – Monograph Series
  • Grenander , U. 1981 . Abstract Inference , New York : Wiley .
  • Gugushvili , S. 2009a . Nonparametric Estimation of the Characteristic Triplet of a Discretely Observed Lévy Process . Journal of Nonparametric Statistics , 21 : 321 – 343 .
  • Gugushvili, S. (2009b), ‘Nonparametric Inference for Discretely Sampled Lévy Processes’, arXiv:0908.3121v2 [math.ST]
  • Gugushvili , S. , Klaassen , C. A.J. and Spreij , P. 2010 . Special Issue: Statistical Inference for Lévy Processes with Applications to Finance , Edited by: Gugushvili , S. , Klaassen , C. A.J. and Spreij , P. 255 – 366 . Oxford, , UK : Wiley-Blackwell . Vol. 64 of Statistica Neerlandica
  • Jacod , J. and Todorov , V. 2010 . Do Price and Volatility Jump Together? . The Annals of Applied Probability , 20 : 1425 – 1469 .
  • Jacod , J. , Klüppelberg , C. and Müller , G. 2011 . “ Testing for Non-Correlation or for a Functional Relationship Between Price and Volatility Jumps ” . submitted for publication
  • Kappus , J. and Reiß , M. 2010 . Estimation of the Characteristics of a Lévy Process Observed at Arbitrary Frequency . Statistica Neerlandica , 64 : 314 – 328 .
  • Kutoyants , Y. A. 1998 . Statistical Inference for Spatial Poisson Processes , New York : Springer .
  • Ledoux , M. 1997 . On Talagrand Deviation Inequalities for Product Measures . ESAIM. Probability and Statistics , 1 : 63 – 87 .
  • Mancini , C. 2005 . Estimating the Integrated Volatility in Stochastic Volatility Models with Lévy Type Jumps http://www.cmap.polytechnique.fr/SEM/finance/2004-2005/mancini1.pdf
  • Massart , P. 2000 . About the Constants in Talagrand's Concentration Inequalities for Empirical Processes . The Annals of Probability , 28 : 863 – 884 .
  • Mazja , W. 1979 . Einbettungssätze für Sobolewsche Räume , Leipzig : Teubner . (Teil 1)
  • Neumann , M. H. and Reiß , M. 2009 . Nonparametric Estimation for Lévy Processes from Low-Frequency Observations . Bernoulli , 15 : 223 – 248 .
  • Reynaud-Bouret , P. 2003 . Adaptive Estimation of the Intensity of Inhomogeneous Poisson Processes via Concentration Inequalities . Probability Theory and Related Fields , 126 : 103 – 153 .
  • Rubin , H. and Tucker , H. G. 1959 . Estimating the Parameters of a Differential Process . Annals of Mathematical Statistics , 30 : 641 – 658 .
  • Rüschendorf , L. and Woerner , J. H.C. 2002 . Expansion of Transition Distributions of Lévy Processes in Small Time . Bernoulli , 8 : 81 – 96 .
  • Sato , K. 1999 . Lévy Processes and Infinitely Divisible Distributions , Cambridge : Cambridge University Press .
  • Shimizu , Y. 2006 . Density Estimation for Lévy Measures for Discretely Observed Diffusion Processes with Jumps . Journal of the Japan Statistical Society , 36 : 37 – 62 .
  • Shimizu , Y. 2009 . Functional Estimation for Lévy Measures of Semimartingales with Poissonian Jumps . Journal of Multivariate Analysis , 100 : 1073 – 1092 .
  • Talagrand , M. 1996 . New Concentration Inequalities in Product Spaces, . Inventiones Mathematicae , 126 : 505 – 563 .
  • van Es , B. , Gugushvili , S. and Spreij , P. 2007 . A Kernel Type Nonparametric Density Estimator for Decompounding . Bernoulli , 13 : 672 – 694 .
  • Watteel , R. N. and Kulperger , R. J. 2003 . Nonparametric Estimation of the Canonical Measure for Infinitely Divisible Distributions . Journal of Statistical Computation and Simulation , 73 : 525 – 542 .
  • Whittaker , E. T. On a New Method of Graduation . Proceedings of the Edinburgh Mathematical Society . Vol. 2 , pp. 63 – 75 .

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