Gassiat, E.; Gautherat, E. Speed of convergence for the blind deconvolution of a linear system with discrete random input. (English) Zbl 0965.62070 Ann. Stat. 27, No. 5, 1684-1705 (1999). Summary: In our paper, IEEE Trans. Inf. Theory 44, No. 5, 1941-1952 (1998; Zbl 0934.94003), we proposed a new estimation method for the blind deconvolution of a linear system with discrete random input, when the observations may be noise perturbed. We give here asymptotic properties of the estimators in the parametric situation. With nonnoisy observations, the speed of convergence is governed by the \(l_1\)-tail of the inverse filter, which may have an exponential decrease. With noisy observations, the estimator satisfies a limit theorem with known distribution, which allows for the construction of confidence regions.To our knowledge, this is the first precise asymptotic result in the noisy blind deconvolution problem with an unknown level of noise. We also extend results concerning Hankel’s estimation to Toeplitz’s estimation and prove a formula to compute Toeplitz forms that may have interest in itself. MSC: 62M09 Non-Markovian processes: estimation 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics Keywords:contrast function; Hankel matrix; discrete linear systems; deconvolution Citations:Zbl 0934.94003 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Anton-Haro, C., Fonollosa, J. A. and Fonollosa, J. R. (1997). Blind channel estimation and data detection using hidden Markov models. IEEE Trans. Signal Process. 45. [2] Bermond, O. and Kéribin, C. (1998). M-Likelihood estimation for noisy MA processes. Unpublished manuscript. [3] Bickel, P. J., Ritov, Y. and Ryden, T. (1998). Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614-1635. · Zbl 0932.62097 · doi:10.1214/aos/1024691255 [4] Cappé, O., Doucet, A., Lavielle, M. and Moulines, E. (1999). Methods for blind maximumlikelihood linear system identification. Signal Process. 73 3-25. · Zbl 0926.93066 · doi:10.1016/S0165-1684(98)00182-0 [5] Gamboa, F. and Gassiat, E. (1996). Blind deconvolution of discrete linear systems. Ann. Statist. 24 1964-1981. · Zbl 0867.62073 · doi:10.1214/aos/1069362305 [6] Gamboa, F. and Gassiat, E. (1997). Source separation when the input sources are discrete or have constant modulus. IEEE Trans. Signal Process. 45 3062-3072. [7] Gassiat, E. and Gautherat, E. (1998). Identification of noisy linear systems with discrete random input. IEEE Trans. Inform. Theory 44 1941-1952. · Zbl 0934.94003 · doi:10.1109/18.705571 [8] Gautherat, E. (1997). Déconvolution aveugle des syst emes linéaires aléatoires bruités ou non. Th ese de l’Université d’Evry-Val d’Essonne. [9] Li, T. H. (1993). Estimation and blind deconvolution of autoregressive systems with nonstationary binary inputs. J. Time Ser. Anal. 14 575-588. · Zbl 0780.62068 · doi:10.1111/j.1467-9892.1993.tb00167.x [10] Li, T. H. (1995). Blind deconvolution of linear systems with multilevel nonstationary inputs. Ann. Statist. 23 690-704. · Zbl 0828.62080 · doi:10.1214/aos/1176324542 [11] Lindsay, B. G. (1989). On the determinants of moment matrices. Ann. Statist. 17 711-721. · Zbl 0672.62062 · doi:10.1214/aos/1176347137 [12] Liu, J. and Chen, R. (1995). Blind deconvolution via sequential imputations. J. Amer. Statist. Assoc. 90 567-576. · Zbl 0826.62062 · doi:10.2307/2291068 [13] Petrov, V. (1975). Sums of Independent Variables. Springer, Berlin. · Zbl 0322.60043 [14] Sashadri, X. (1994). Joint data and channel estimation using blind trellis search techniques. In Blind Deconvolution (S. Hawking, ed.) 259-286. Prentice Hall, Englewood Cliffs, NJ. [15] van der Veen, A. J., Talwar, S. and Paulraj, A. (1997). A subspace approach to blind space-time signal processing for wireless communication systems. IEEE Trans. Signal Process. 45 173-190. 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.