On the convergence rate for the estimation of impulse response function in the space Lp(T)
DOI:
https://doi.org/10.17721/1812-5409.2018/4.5Abstract
The problem of estimation of a stochastic linear system has been a matter of active research for the last years. One of the simplest models considers a ‘black box’ with some input and a certain output. The input may be single or multiple and there is the same choice for the output. This generates a great amount of models that can be considered. The sphere of applications of these models is very extensive, ranging from signal processing and automatic control to econometrics (errors-in-variables models). In this paper a time-invariant continuous linear system is considered with a real-valued impulse response function. We assume that impulse function is square-integrable. Input signal is supposed to be Gaussian stationary stochastic process with known spectral density. A sample input–output cross-correlogram is taken as an estimator of the response function. An upper bound for the tail of the distribution of the estimation error is found that gives a convergence rate of estimator to impulse response function in the space Lp(T).
Key words: impulse response function, linear time-invariant system (LTI), Gaussian process, cross-correlogram.
Pages of the article in the issue: 36-41
Language of the article: Ukrainian
References
BULDYGIN V., BLAZHIEVSKA I. (2010) Asymptotic properties of cross-correlogram estimators of impulse response functions in linear system Research // Bulletin of National Technical University of Ukraine "KPI", 4, 16–27.
BULDYGIN V., KUROTSCHKA V. (1999) On cross-coorrelogram estimators of the response function in continuous linear systems from discrete observations // Random Oper. and Stoch. Equ., 7, №1, 71–90.
BULDYGIN V., FU LI (1997) On asymptotic normality of an estimation of unit impulse responses of linear system I, II // Theor. Probability and Math. Statist., 54, 55, 3–17, 30–37.
BULDYGIN V., UTZET F., ZAIATS V. (2004) Asymptotic normality of crosscorrelogram estimators of the response function // Statistical Infernce for Stochastic Processes, 7, 1–34.
GIKHMAN I, SKOROKHOD, A. (1996) Introduction to the Theory of Random Processes, Dover Publication, 544 p.
KOZACHENKO YU., PASHKO A., ROZORA I. (2007) Simulation of Stochastic Processes and fields, Zadruga, Kyiv. (in Ukrainian)
KOZACHENKO YU., ROZORA I. (2015) On cross-correlogram estimators of impulse response functions // Theor. Probability and Math. Statist., 93, 75-86.
KOZACHENKO YU., TROSHKI V. (2014) A criterion for testing hypotheses about the covariance function of a stationary Gaussian stochastic process // Modern Stochastics: Theory and Applications, 1, 139–149.
KOZACHENKO YU., STUS O. (1998) Square-Gaussian random processes and estimators of covariance functions // Math. Communications, 3, №1, 83–94.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
