Abstract
We use moderate deviations to study the signal detection problem for a diffusion model. We establish a moderate deviation principle for the log-likelihood function of the diffusion model. Then applying the moderate deviation estimates to hypothesis testing for signal detection problem we give a decision region such that its error probability of the second kind tends to zero with faster speed than the error probability of the first kind when the error probability of the first kind is approximated by e−αr(T), where α > 0, r(T) = o(T) and r(T)→∞ as the observation time T goes to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blahut R E. Hypothesis testing and information theory. IEEE Trans Inform Theory, 1974, 20: 405–415
Chiyonobu T. Hypothesis testing for signal detection problem and large deviations. Nagoya Math J, 2003, 162: 187–203
Dembo A, Zeitouni O. Large Deviation Technique and Applicatons. New York: Springer, 1998
Deuschel J D, Stroock D W. Large Deviations. London: Academic Press, 1989
Fitzsimmons P J. Absolute continuity of symmetric diffusion. Ann Probab, 1997, 25: 230–258
Fukushima M, Oshima Y, Takeda Y M. Dirichlet Forms and Symmetric Markov Processes. Berlin: Walter de Gruyter, 1994
Gao F Q. Small perturbation cramer methods and moderate deviations for Markov processes. Acta Math Scientia, 1995, 15: 394–405
Gao F Q. Moderate deviations for martingales and mixing random processes. Stoch Proce Appl, 1996, 61: 263–275
Glynn P W, Ormoneit D. Hoeffding’s inequality for uniformly ergodic Markov chains. Stat Probab Lett, 2002, 56: 143–146
Han T S, Kobayashi K. The strong converse theorem in hypothesis testing. IEEE Trans Inform Theory, 1989, 35: 178–180
Ihara S, Sakuma Y. Signal detection in white Gaussian channel. In: Proc Seventh Japan-Russian Sym Probab Theory, Math Stat. Singapore: World Scientific, 1996, 147–156
Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Kodansha: North-Holland, 1981
Mogulskii A A. On moderately large deviations from the invariant measure. In: Borovkov A A, ed. Advance in Probability Theory: Limit Theorems and Related Problem. New York: Optimization Software Inc, 1984, 163–175
Nakagawa K, Kanaya F. On the converse theorem in statistical hypothesis testing for Markov chains. IEEE Trans Inform Theory, 1993, 39: 629–633
Wang F Y. Sharp Explicit lower bounds of heat kernels. Ann Probab, 1997, 25: 1995–2006
Wu L M. Moderate deviations of dependent random variables related to CLT. Ann Probab, 1995, 23: 420–445
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gao, F., Zhao, S. Moderate deviations and hypothesis testing for signal detection problem. Sci. China Math. 55, 2273–2284 (2012). https://doi.org/10.1007/s11425-012-4514-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-012-4514-8