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

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **92** (2015).

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

Full-text PDF

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.

**1.**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.**2.**L. Chaumont and G. Uribe Bravo,*Markovian bridges: Weak continuity and pathwise constructions*, Ann. Probab.**39(2)**(2011), 609-647. MR**2789508****3.**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****4.**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****5.**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****6.**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****7.**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****8.**L. Le Cam and G. L. Yang,*Asymptotics in Statistics*, Springer, Berlin-New York, 1990. MR**1066869****9.**H. Masuda,*Ergodicity and exponential -mixing bounds for multidimensional diffusions with jumps*, Stoch. Proc. Appl.**117**(2007), 35-56. MR**2287102****10.**I. I. Gihman and A. V. Skorohod,*Stochastic Differential Equations and their Applications*, ``Naukova dumka'', Kiev, 1982. (Russian) MR**0263172****11.**A. N. Shiryaev,*Probability*, MCNMO, Moscow, 2004; English transl. of the first Russian (1980) edition, Springer-Verlag, Berlin-Heidelberg-New York, 1996. MR**1368405**

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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

DOI:
https://doi.org/10.1090/tpms/978

Keywords:
Asymptotic efficiency,
local asymptotic normality,
L\'evy processes,
stochastic differential equations

Received by editor(s):
May 21, 2015

Published electronically:
August 10, 2016

Article copyright:
© Copyright 2016
American Mathematical Society