Asymptotic properties of the least squares estimators of a two dimensional model

Abstract. We consider a particular two dimensional model, which has a wide applications in statistical signal processing and texture classifications. We prove the consistency of the least squares estimators of the model parameters and also obtain the asymptotic distribution of the least squares estimators. We observe the strong consistency of the least squares estimators when the errors are independent and identically distributed double array random variables. We show that the asymptotic distribution of the least squares estimators are multivariate normal. It is observed that the asymptotic dispersion matrix coincides with the Cramer-Rao lower bound. This paper generalizes some of the existing one dimensional results to the two dimensional case. Some numerical experiments are performed to see how the asymptotic results work for finite samples.

[1]  Changbao Wu,et al.  Asymptotic Theory of Nonlinear Least Squares Estimation , 1981 .

[2]  William H. Press,et al.  Numerical Recipes in Fortran 77 , 1992 .

[3]  James H. McClellan,et al.  The extension of Pisarenko's method to multiple dimensions , 1982, ICASSP.

[4]  Kai Lai Chung,et al.  A Course in Probability Theory , 1949 .

[5]  D. Kundu.  Asymptotic theory of least squares estimator of a particular nonlinear regression model , 1993 .

[6]  Yingbo Hua.  Estimating two-dimensional frequencies by matrix enhancement and matrix pencil , 1992, IEEE Trans. Signal Process..

[7]  O. Bunke,et al.  Nonlinear regression, functional relations and robust methods , 1990 .

[8]  N. K. Bose,et al.  Parameter Estimation via Signal-Selectivity of Signal-Subspace (Pess) and Its Application To 2-D Wavenumber Estimation , 1995 .

[9]  A. M. Walker.  On the estimation of a harmonic component in a time series with stationary independent residuals , 1971 .

[10]  D. Kundu,et al.  Asymptotic theory of least squares estimator of a nonlinear time series regression model , 1996 .

[11]  Visvaldis Mangulis,et al.  Handbook of series for scientists and engineers , 1965 .

[12]  T. Hasan.  NONLINEAR TIME SERIES REGRESSION FOR A CLASS OF AMPLITUDE MODULATED CONSINUSOIDS , 1982 .

[13]  Piero Barone,et al.  A two-dimensional Prony's method for spectral estimation , 1992, IEEE Trans. Signal Process..

[14]  R. Jennrich.  Asymptotic Properties of Non-Linear Least Squares Estimators , 1969 .

[15]  T. Lai,et al.  A Law of the Iterated Logarithm for Double Arrays of Independent Random Variables with Applications to Regression and Time Series Models , 1982 .

[16]  Yingbo Hua,et al.  Estimating two-dimensional frequencies by matrix enhancement and matrix pencil , 1991, [Proceedings] ICASSP 91: 1991 International Conference on Acoustics, Speech, and Signal Processing.

[17]  Raymond J. Carroll,et al.  Nonlinear Regression, Functional Relations and Robust Methods. , 1990 .