The Annals of Statistics

Asymptotically optimal parameter estimation under communication constraints

Georgios Fellouris

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 40, Number 4 (2012), 2239-2265.

Dates
First available in Project Euclid: 23 January 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1358951381

Digital Object Identifier
doi:10.1214/12-AOS1035

Mathematical Reviews number (MathSciNet)
MR3059082

Zentralblatt MATH identifier
1297.62182

Subjects
Primary: 62L12: Sequential estimation 62F30: Inference under constraints
Secondary: 62F12: Asymptotic properties of estimators 62M05: Markov processes: estimation 62M09: Non-Markovian processes: estimation

Keywords
Asymptotic optimality communication constraints decentralized estimation quantization random sampling sequential estimation semimartingale

Citation

Fellouris, Georgios. Asymptotically optimal parameter estimation under communication constraints. Ann. Statist. 40 (2012), no. 4, 2239--2265. doi:10.1214/12-AOS1035. https://projecteuclid.org/euclid.aos/1358951381


Export citation

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.
  • [3] Brown, B. M. and Hewitt, J. I. (1975). Inference for the diffusion branching process. J. Appl. Probab. 12 588–594.
  • [4] Feigin, P. D. (1976). Maximum likelihood estimation for continuous-time stochastic processes. Adv. in Appl. Probab. 8 712–736.
  • [5] Fellouris, G. and Moustakides, G. V. (2011). Decentralized sequential hypothesis testing using asynchronous communication. IEEE Trans. Inform. Theory 57 534–548.
  • [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.
  • [8] Grenander, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195–277.
  • [9] Han, T. S. and Amari, S. (1995). Parameter estimation with multiterminal data compression. IEEE Trans. Inform. Theory 41 1802–1833.
  • [10] Han, T. S. and Amari, S. (1998). Statistical inference under multiterminal data compression. IEEE Trans. Inform. Theory 44 2300–2324.
  • [11] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [12] Kutoyants, Y. A. (2004). Statistical Inference for Ergodic Diffusion Processes. Springer, London.
  • [13] Liptser, R. S. and Shiryaev, A. N. (2001). Statistics of Random Processes: Applications, 2nd ed. Applications of Mathematics (New York) 6. Springer, Berlin.
  • [14] Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41 520–527.
  • [15] Luo, Z.-Q. (2005). Universal decentralized estimation in a bandwidth constrained sensor network. IEEE Trans. Inform. Theory 51 2210–2219.
  • [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.
  • [17] Mel’nikov, A. V. and Novikov, A. A. (1988). Sequential inferences with guaranteed accuracy for semimartingales. Teor. Veroyatn. Primen. 33 480–494.
  • [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.
  • [20] Rabi, M., Moustakides, G. V. and Baras, J. S. (2012). Adaptive sampling for linear state estimation. SIAM J. Control Optim. 50 672–702.
  • [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.
  • [22] Striebel, C. T. (1959). Densities for stochastic processes. Ann. Math. Statist. 30 559–567.
  • [23] Veeravalli, V. V. (1999). Sequential decision fusion: Theory and applications. J. Franklin Inst. 336 301–322.
  • [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.