Asymptotic properties of the method of empirical mean for stationary random processes and homogeneous random fields

DOI: 10.31673/2412-4338.2019.014654

Abstract

The article considers the quality of empirical estimates of unknown parameters of stationary random processes and homogeneous random fields for which the conditions of ergodicity or strong mixing are satisfied. A series of statements on consistency of estimates, asymptotic distribution and large deviations for estimations of unknown parameter obtained by the method of empirical means for independent or weakly dependent observations was formulated.

Keywords: method of empirical mean, asymptotic properties, consistency, estimate, large deviations, criterion function.

References

1. Knopov P. S., Kasitskaya E. J. Empirical Estimates in Stochastic Optimization and Identification. “Kluwer Academic Publishers”, Dordrecht, 2002.
2. Knopov P. S. On Some Classes of M-estimates for Non-stationary Regression Models, Theory of Stochastic Processes 3 (19) (1997), No. 1-2, 222–238.
3. Gololobov D.A., Kasitskaya E. J. Asymptotic properties of the method of observed mean for homogeneous random fields, Cybernetics and Systems Analysis 49 (2013), No. 3, 465–471.
4. Knopov P. S., Korkhin A.S. Regression Analysis Under A Priori Parameter Restrictions, “Springer”, New York, 2012.
5. Deuschel J. D., Strook D.W.. Large Deviations, “Academic Press”, Boston, 1989.
6. Knopov P. S., Kasitskaya E. J. On large deviations of empirical estimates in stochastic programming problems, Cybernetics and Systems Analysis 40 (2004), No. 4, 510–516.
7. Ermoliev Yu.M., Wets R.J-B. Numerical techniques for stochastic optimization, Springer series in computational mathematics. Vol. 10. Berlin : Springer-Verlag, 1988.
8. Le Cam L. On some asymptotic properties of maximum likelihood estimates and related Bayes estimates, California, Publ. Statist, 1953, Vol. 1, No. 11, 277–330.
9. Pfanzagl J. On the measurability and consistency of minimum contrast estimates. Metrika 14 (1969), Issue 1, 249–272.
10. Knopov P. S., Kasitskaya E. J. Properties of empirical estimates in stochastic optimization and identification problems, Annals of Operations Research 56 (1995), Issue 1, 225–239.
11. Knopov P. S., Norkin V. I. Convergence Conditions for the Observed Mean Method in Stochastic Programming, Cybernetics and Systems Analysis 54 (2018), Issue 1, 45–59.