Teo Sharia (2003) On Recursive Parametric Estimation Theory.
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The classical non-recursive methods to estimate unknown parameters of the model, such as the maximum likelihood method, the method of least squares etc. eventually require maximization procedures. These methods are often difficult to implement, especially for non i.i.d. models. If for every sample size n, when new data are acquired, an estimator has to be computed afresh, and if a numerical method is needed to do so, it generally becomes very laborious. Therefore, it is important to consider recursive estimation procedures which are appealing from the computational point of view. Recursive procedures are those which at each step allow one to re-estimate values of unknown parameters based on the values already obtained at the previous step together with new information. We propose a wide class of recursive estimation procedures for the general statistical model and study convergence, the rate of convergence, and the local asymptotic linearity. Also, we demonstrate the use of the results on some examples.
This is a Published version This version's date is: 24/01/2003 This item is peer reviewed
https://repository.royalholloway.ac.uk/items/40c3513f-1083-9145-a9d7-3d1adf506140/1/
Deposited by () on 14-Jul-2010 in Royal Holloway Research Online.Last modified on 10-Dec-2010
Anderson, T.W. (1959). On asymptotic distributions of estimates of parametersof stochastic dierence equations. Ann. Math. Statist. 30,676{687.Barndorff-Nielsen, O.E. (1988). Parametric Statistical Models andLikelihood. Springer Lecture Notes in Statistics 50. Heidelberg, Springer.Barndorff-Nielsen, O.E. and Sorensen, M. (1994). A review of someaspects of asymptotic likelihood theory for stochastic processes. Inter-national Statistical Review. 62, 1, 133-165.Basawa, I.V. and Scott, D.J. (1983). Asymptotic Optimal Inference forNon-ergodic Models. Springer-Verlag, New York.Campbell, K. (1982). Recursive computation of M-estimates for the parametersof a nite autoregressive process. Ann. Statist. 10, 442-453.Denby, L. and Martin, R.D. (1979). Robust estimation of the rst orderautoregressive parameter. J. Amer. Statist. Assoc. 74, 140-146.Englund, J.-E., Holst, U., and Ruppert, D. (1989) Recursive estimatorsfor stationary, strong mixing processes { a representation theorem andasymptotic distributions Stochastic Processes Appl. 31, 203{222.Fabian, V. (1978). On asymptotically ecient recursive estimation, Ann.Statist. 6, 854-867.Feigin, P.D. (1981). conditional exponential families and a representationtheorem for asymptotic inference. Ann. Statist. 9, 597-603.Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., and Stahel, W.(1986). Robust Statistics - The Approach Based on Inuence Functions.Wiley, New YorkHuber, P.J. (1981). Robust Statistics. Wiley, New York.Jureckova, J. and Sen, P.K. (1996). Robust Statistical Procedures -Asymptotics and Interrelations. Wiley, New York.Khas'minskii, R.Z. and Nevelson, M.B. (1972). Stochastic Approxima-tion and Recursive Estimation. Nauka, Moscow.Launer, R.L. and Wilkinson, G.N. (1979). Robustness in Statistics.Academic Press, New York.Lazrieva, N., Sharia, T. and Toronjadze, T. (1997). The Robbins-Monro type stochastic dierential equations. I. Convergence of solutions.Stochastics and Stochastic Reports 61, 67{87.Lazrieva, N., Sharia, T. and Toronjadze, T. (2003). The Robbins-Monro type stochastic dierential equations. II. Asymptotic behaviourof solutions. Stochastics and Stochastic Reports (in print).Lazrieva, N. and Toronjadze, T. (1987). Ito-Ventzel's formula for semimartingales,asymptotic properties of MLE and recursive estimation.Lect. Notes in Control and Inform. Sciences, 96, Stochast. di. sys-tems, H.J, Engelbert, W. Schmidt (Eds.), Springer 346{355.Lehman, E.L. (1983). Theory of Point Estimation. Wiley, New York.Leonov, S.L. (1988). On recurrent estimation of autoregression parameters,Avtomatika i Telemekhanika 5, 105-116.Liptser, R.S. and Shiryayev, A.N. (1989). Theory of Martingales. Kluwer,Dordrecht.Ljung, L. Pflug, G. and Walk, H. (1992). Stochastic Approximationand Optimization of Random Systems. Birkhauser, Basel.Prakasa Rao, B.L.S. (1999). Semimartingales and their Statistical Infer-ence. Chapman & Hall, New York.Rieder, H. (1994). Robust Asymptotic Statistics. Springer{Verlag, NewYork.Robbins, H. Monro, S. (1951) A stochastic approximation method, Ann.Statist. 22, 400{407.Robbins, Siegmund, H.D. (1971) A convergence theorem for nonnegativealmost supermartingales and some applications, Optimizing Methodsin Statistics, ed. J.S. Rustagi Academic Press, New York. 233{257.Serfling, R.J. (1980). Approximation Theorems of Mathematical Statis-tics. Wiley, New York.Sharia, T. (1992). On the recursive parameter estimation for general statisticalmodel in discrete time. Bulletin of the Georgian Acad. Scienc.145 3, 465{468.Sharia, T. (1998). On the recursive parameter estimation for the generaldiscrete time statistical model. Stochastic Processes Appl. 73, 2, 151{172.Sharia, T. (1997). Truncated recursive estimation procedures. Proc. A.Razmadze Math. Inst. 115, 149{159. Shiryayev, A.N. (1984). Probability. Springer-Verlag, New York.Titterington, D.M., Smith, A.F.M. and Makov, U.E. (1985). Statistical Analysis of Finite Mixture Distributions. John Wiley & Sons, New York.White, J.S. (1958). The limiting distribution of the serial correlation coecient in the explosive case. Ann. Math. Stat. 29, 1188{1197.