×

Asymptotically optimal parameter estimation under communication constraints. (English) Zbl 1297.62182

Summary: A parameter estimation problem is considered, in which dispersed sensors transmit to the statistician partial information regarding their observations. The sensors observe the paths of continuous semimartingales, whose drifts are linear with respect to a common parameter. A novel estimating scheme is suggested, according to which each sensor transmits only one-bit messages at stopping times of its local filtration. The proposed estimator is shown to be consistent and, for a large class of processes, asymptotically optimal, in the sense that its asymptotic distribution is the same as the exact distribution of the optimal estimator that has full access to the sensor observations. These properties are established under an asymptotically low rate of communication between the sensors and the statistician. Thus, despite being asymptotically efficient, the proposed estimator requires minimal transmission activity, which is a desirable property in many applications. Finally, the case of discrete sampling at the sensors is studied when their underlying processes are independent Brownian motions.

MSC:

62L12 Sequential estimation
62F30 Parametric inference under constraints
62F12 Asymptotic properties of parametric estimators
62M05 Markov processes: estimation; hidden Markov models
62M09 Non-Markovian processes: estimation
60G44 Martingales with continuous parameter
PDFBibTeX XMLCite
Full Text: DOI arXiv Euclid

References:

[1] Blum, R. S., Kassam, S. A. and Poor, H. V. (1997). Distributed detection with multiple sensors: Part II-advanced topics. Proc. IEEE 85 64-79.
[2] Brown, B. M. and Hewitt, J. I. (1975). Asymptotic likelihood theory for diffusion processes. J. Appl. Probab. 12 228-238. · Zbl 0314.62036 · doi:10.2307/3212436
[3] Brown, B. M. and Hewitt, J. I. (1975). Inference for the diffusion branching process. J. Appl. Probab. 12 588-594. · Zbl 0313.60058 · doi:10.2307/3212875
[4] Feigin, P. D. (1976). Maximum likelihood estimation for continuous-time stochastic processes. Adv. in Appl. Probab. 8 712-736. · Zbl 0355.62086 · doi:10.2307/1425931
[5] Fellouris, G. and Moustakides, G. V. (2011). Decentralized sequential hypothesis testing using asynchronous communication. IEEE Trans. Inform. Theory 57 534-548. · Zbl 1366.94141 · doi:10.1109/TIT.2010.2090249
[6] Foresti, G. L., Regazzoni, C. S. and Varshney, P. K., eds. (2003). Multisensor Surveillance Systems : The Fusion Perspective . Kluwer Academic, Dordrecht.
[7] Galtchouk, L. and Konev, V. (2001). On sequential estimation of parameters in semimartingale regression models with continuous time parameter. Ann. Statist. 29 1508-1536. · Zbl 1043.62067 · doi:10.1214/aos/1013203463
[8] Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195-277. · Zbl 0058.35501 · doi:10.1007/BF02590638
[9] Han, T. S. and Amari, S. (1995). Parameter estimation with multiterminal data compression. IEEE Trans. Inform. Theory 41 1802-1833. · Zbl 0844.62005 · doi:10.1109/18.476308
[10] Han, T. S. and Amari, S. (1998). Statistical inference under multiterminal data compression. IEEE Trans. Inform. Theory 44 2300-2324. · Zbl 0933.94014 · doi:10.1109/18.720540
[11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus , 2nd ed. Graduate Texts in Mathematics 113 . Springer, New York. · Zbl 0734.60060
[12] Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes . Springer, London. · Zbl 1038.62073
[13] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes : Applications , 2nd ed. Applications of Mathematics ( New York ) 6 . Springer, Berlin. · Zbl 0369.60001
[14] Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41 520-527. · Zbl 0212.49703 · doi:10.1214/aoms/1177697092
[15] Luo, Z.-Q. (2005). Universal decentralized estimation in a bandwidth constrained sensor network. IEEE Trans. Inform. Theory 51 2210-2219. · Zbl 1309.94046 · doi:10.1109/TIT.2005.847692
[16] Martinsek, A. T. (1981). A note on the variance and higher central moments of the stopping time of an SPRT. J. Amer. Statist. Assoc. 76 701-703. · Zbl 0473.62072 · doi:10.2307/2287533
[17] Mel’nikov, A. V. and Novikov, A. A. (1988). Sequential inferences with guaranteed accuracy for semimartingales. Teor. Veroyatn. Primen. 33 480-494. · Zbl 0693.62066 · doi:10.1137/1133071
[18] Novikov, A. A. (1972). Sequential estimation of the parameters of processes of diffusion type. Mat. Zametki 12 627-638.
[19] Prakasa Rao, B. L. S. (1985). Statistical Inference for Diffusion Type Processes . Arnold, London. · Zbl 0952.62077
[20] Rabi, M., Moustakides, G. V. and Baras, J. S. (2012). Adaptive sampling for linear state estimation. SIAM J. Control Optim. 50 672-702. · Zbl 1318.62268 · doi:10.1137/090757125
[21] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion , 3rd ed. Grundlehren der Mathematischen Wissenschaften [ Fundamental Principles of Mathematical Sciences ] 293 . Springer, Berlin. · Zbl 0917.60006
[22] Striebel, C. T. (1959). Densities for stochastic processes. Ann. Math. Statist. 30 559-567. · Zbl 0091.13501 · doi:10.1214/aoms/1177706268
[23] Veeravalli, V. V. (1999). Sequential decision fusion: Theory and applications. J. Franklin Inst. 336 301-322. · Zbl 1048.90125 · doi:10.1016/S0016-0032(98)00023-4
[24] Viswanathan, R. and Varshney, R. K. (1997). Distributed detection with multiple sensors: Part II-fundamentals. Proc. IEEE 85 54-63.
[25] Xiao, J.-J. and Luo, Z.-Q. (2005). Decentralized estimation in an inhomogeneous sensing environment. IEEE Trans. Inform. Theory 51 3564-3575. · Zbl 1320.94030 · doi:10.1109/TIT.2005.855580
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.