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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

Estimation for the discretely observed telegraph process


Authors: S. M. Iacus and N. Yoshida
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 78 (2008).
Journal: Theor. Probability and Math. Statist. 78 (2009), 37-47
MSC (2000): Primary 60K99; Secondary 62M99
Published electronically: August 4, 2009
MathSciNet review: 2446847
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Abstract: The telegraph process $ \{X(t), t>0\}$ is supposed to be observed at $ n+1$ equidistant time points $ t_i=i\Delta_n$, $ i=0,1,\dots, n$. The unknown value of $ \lambda$, the underlying rate of the Poisson process, is a parameter to be estimated. The asymptotic framework considered is the following: $ \Delta_n \to 0$, $ n\Delta_n=T \to \infty$ as $ n \to \infty$. We show that previously proposed moment type estimators are consistent and asymptotically normal but not efficient. We study further an approximated moment type estimator which is still not efficient but comes in explicit form. For this estimator the additional assumption $ n\Delta_n^3 \to 0$ is required in order to obtain asymptotic normality. Finally, we propose a new estimator which is consistent, asymptotically normal and asymptotically efficient under no additional hypotheses.


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Additional Information

S. M. Iacus
Affiliation: Department of Economics, Business and Statistics, University of Milan, Via Conservatorio 7, 20122 Milan, Italy
Email: stefano.iacus@unimi.it

N. Yoshida
Affiliation: Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguroku, Tokyo 153-8914, Japan
Email: nakahiro@ms.u-tokyo.ac.jp

DOI: http://dx.doi.org/10.1090/S0094-9000-09-00760-1
PII: S 0094-9000(09)00760-1
Keywords: Telegraph process, discretely observed process, inference for stochastic processes
Received by editor(s): December 28, 2006
Published electronically: August 4, 2009
Additional Notes: The work of the first author was supported by JSPS (Japan Society for the Promotion of Science) Program FY2006, grant ID No. S06174. He is also thankful to the Graduate School of Mathematical Sciences, University of Tokyo as host research institute for the JSPS Program
Article copyright: © Copyright 2009 American Mathematical Society