A method for checking efficiency of estimators in statistical models driven by Lévy’s noise
Authors:
S. V. Bodnarchuk and D. O. Ivanenko
Translated by:
N. Semenov
Journal:
Theor. Probability and Math. Statist. 92 (2016), 1-15
MSC (2010):
Primary 62F12; Secondary 60G51
DOI:
https://doi.org/10.1090/tpms/978
Published electronically:
August 10, 2016
MathSciNet review:
3553422
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: A method for checking the efficiency of estimators of unknown parameters is proposed for statistical models with observations described by a stochastic differential equation driven by Lévy’s noise.
References
- S. V. Bodnarchuk and A. M. Kulik, Stochastic control based on time-change transformations for stochastic processes with Lévy noise, Teor. Ĭmovir. Mat. Stat. 86 (2012), 11–27; English transl. in Theor. Probability and Math. Statist. 86 (2013), 13–31.
- Loïc Chaumont and Gerónimo Uribe Bravo, Markovian bridges: weak continuity and pathwise constructions, Ann. Probab. 39 (2011), no. 2, 609–647. MR 2789508, DOI 10.1214/10-AOP562
- Jaroslav Hájek, Local asymptotic minimax and admissibility in estimation, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 175–194. MR 0400513
- D. O. Ivanenko and A. M. Kulik, Malliavin calculus approach to statistical inference for Lévy driven SDE’s, Methodol. Comput. Appl. Probab. 17 (2015), no. 1, 107–123. MR 3306674, DOI 10.1007/s11009-013-9387-y
- D. Ivanenko and A. Kulik, LAN property for discretely observed solutions to Lévy driven SDE’s, Mod. Stoch. Theory Appl. 1 (2014), no. 1, 33–47. MR 3314792, DOI 10.15559/vmsta-2014.1.1.4
- Alexey M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stochastic Process. Appl. 119 (2009), no. 2, 602–632. MR 2494006, DOI 10.1016/j.spa.2008.02.006
- L. Le Cam, Limits of experiments, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, Calif., 1970/1971) Univ. California Press, Berkeley, Calif., 1972, pp. 245–261. MR 0415819
- Lucien Le Cam and Grace Lo Yang, Asymptotics in statistics, Springer Series in Statistics, Springer-Verlag, New York, 1990. Some basic concepts. MR 1066869, DOI 10.1007/978-1-4684-0377-0
- Hiroki Masuda, Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps, Stochastic Process. Appl. 117 (2007), no. 1, 35–56. MR 2287102, DOI 10.1016/j.spa.2006.04.010
- I. I. Gihman and A. V. Skorohod, Stokhasticheskie differentsial′nye uravneniya, Izdat. “Naukova Dumka”, Kiev, 1968 (Russian). MR 0263172
- A. N. Shiryaev, Probability, 2nd ed., Graduate Texts in Mathematics, vol. 95, Springer-Verlag, New York, 1996. Translated from the first (1980) Russian edition by R. P. Boas. MR 1368405, DOI 10.1007/978-1-4757-2539-1
References
- S. V. Bodnarchuk and A. M. Kulik, Stochastic control based on time-change transformations for stochastic processes with Lévy noise, Teor. Ĭmovir. Mat. Stat. 86 (2012), 11–27; English transl. in Theor. Probability and Math. Statist. 86 (2013), 13–31.
- L. Chaumont and G. Uribe Bravo, Markovian bridges: Weak continuity and pathwise constructions, Ann. Probab. 39(2) (2011), 609–647. MR 2789508
- J. Hájek, Local asymptotic minimax admissibility in estimation, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley–Los Angeles, 1971, pp. 175–194. MR 0400513
- D. O. Ivanenko and A. M. Kulik, Malliavin calculus approach to statistical inference for Lévy driven SDE’s, Methodol. Comput. Appl. Probab. (2013). MR 3306674
- D. O. Ivanenko and A. M. Kulik, LAN property for discretely observed solutions to Lévy driven SDE’s, Modern Stochastics: Theory and Appl. 1 (2014), 33–47. MR 3314792
- A. M. Kulik, Exponential ergodicity of the solutions to SDE’s with a jump noise, Stoch. Process. Appl. 119 (2009), no. 2, 602–632. MR 2494006
- L. Le Cam, Limits of experiments, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley–Los Angeles, 1971, pp. 245–261, MR 0415819
- L. Le Cam and G. L. Yang, Asymptotics in Statistics, Springer, Berlin–New York, 1990. MR 1066869
- H. Masuda, Ergodicity and exponential $\beta$-mixing bounds for multidimensional diffusions with jumps, Stoch. Proc. Appl. 117 (2007), 35–56. MR 2287102
- I. I. Gihman and A. V. Skorohod, Stochastic Differential Equations and their Applications, “Naukova dumka”, Kiev, 1982. (Russian) MR 0263172
- A. N. Shiryaev, Probability, MCNMO, Moscow, 2004; English transl. of the first Russian (1980) edition, Springer-Verlag, Berlin–Heidelberg–New York, 1996. MR 1368405
Similar Articles
Retrieve articles in Theory of Probability and Mathematical Statistics
with MSC (2010):
62F12,
60G51
Retrieve articles in all journals
with MSC (2010):
62F12,
60G51
Additional Information
S. V. Bodnarchuk
Affiliation:
Department of Mathematical Analysis and Probability Theory, National Technical University of Ukraine “KPI”, Peremogy Avenue 37, 03056, Kyiv, Ukraine
D. O. Ivanenko
Affiliation:
Department of Mathematics and Theoretical Radiophysics, Faculty for Radiophysics, Electronics, and Computer Systems, Taras Shevchenko National University of Kyiv, Academician Glushkov Avenue, 4, Kyiv 03127, Ukraine
Email:
ida@univ.net.ua
Keywords:
Asymptotic efficiency,
local asymptotic normality,
Lévy processes,
stochastic differential equations
Received by editor(s):
May 21, 2015
Published electronically:
August 10, 2016
Article copyright:
© Copyright 2016
American Mathematical Society
