Abstract
To estimate the unknown autoregression parameter in the case when noise has an infinite dispersion, the weighted estimate by the least-squares method is suggested. The limit distribution of the error of estimation is obtained. It is shown that the weighted estimate is asymptotically more exact in comparison with the common estimate by the least-squares method.
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References
Ershov, A.A., Stable Estimation Methods of Parameters (Survey), Avtom. Telemekh., 1978, no. 8, pp. 66–100.
Resnick, S., Special Invited Paper: Heavy Tail Modeling and Teletraffic Data, Ann. Statist., 1997, vol. 25, no. 5, pp. 1805–1869.
Fallot, P. and Le Breton, A., On the Stochastic Linear Regulator Problem for Systems with Infinite Variance, Syst. Control Lett., 1993, no. 21, pp. 423–429.
Zolotarev, V.M., Odnomernye ustoichivye raspredeleniya (Univariate Stable Distributions), Moscow: Nauka, 1983.
Ljung, L. and Soderstrom, T., Theory and Practice of Recursive Identification, London: MIT Press, 1983.
Goryainova, E.R., Sign Estimates in the Model of Moving Average with Infinite Dispersion, Avtom. Telemekh., 1997, no. 11, pp. 29–38.
Davis, R.A. and Resnick, S., Limit Theory for the Sample Covariance and Correlation Functions of Moving Averages, Ann. Statist., 1986, vol. 14, no. 2, pp. 533–558.
Knight, K., Rate of Convergence of Centered Estimates of Autoregressive Parameters for Infinite Variance Autoregressions, J. Ser. Anal., 1987, vol. 8, no. 1, pp. 51–60.
Vasil’ev, V.A., Dobrovidov, A.V., and Koshkin, G.V., Neparametricheskoe otsenivanie funktsionalov ot raspredelenii statsionarnykh posledovatel’nostei (Nonparametric Estimation of Functionals from Distributions of Stationary Sequences), Moscow: Nauka, 2004.
Feller, V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya (Introduction to Theory of Probabilities and Its Applications), Moscow: Mir, 1984, vol. 2.
Breiman, L., On Some Limit Theorems Similar to Arc-Sin Law, Theory Probab. Appl., 1965, vol. 10, pp. 323–331.
Davis, R.A., Stable Limits for Partial Sums of Dependent Random Variables, Ann. Prob., 1983, vol. 11, no. 2, pp. 262–269.
Ibragimov, I.A. and Linnik, Yu.V., Nezavisimye i statsionarno svyazannye velichiny (Independent and Stationary Related Quantities), Moscow: Nauka, 1965.
Billingsley, P., Skhodimost’ veroyatnostnykh mer (Convergence of Probability Measures), Moscow: Nauka, 1977.
Shiryaev, A.N., Veroyatnost’ (Probability), Moscow: Nauka, 1980.
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Original Russian Text © A.S. Markov, 2009, published in Avtomatika i Telemekhanika, 2009, No. 1, pp. 104–118.
This work was supported by the Russian Foundation for Basic Research, project no. 04-01-00855.
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Markov, A.S. Estimation of the autoregression parameter with infinite dispersion of noise. Autom Remote Control 70, 92–106 (2009). https://doi.org/10.1134/S000511790901007X
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DOI: https://doi.org/10.1134/S000511790901007X