Parameter estimation for a stationary process on a d-dimensional lattice

On estimation of parameters for spatial autoregressive model

“…There are several types of estimators of the vector of parameters a, most common are Least Squares (LS), Yule-Walker (YW) type, and Maximum likelihood (ML) estimators. In Tjostheim (1978Tjostheim ( , 1983 was stated that LS and YW estimates are asymptotically the same and for small samples and strongly correlated series YW estimates can be considerably more biased (see also Tjostheim and Paulsen (1983) and Guyon (1982)). Recently Basu and Reinsel claimed (see Basu and Reinsel 1992) that in Tjostheim (1978) in the proof there is an error and Gaussian asymptotic distribution of YW estimator contains an asymptotic bias term.…”

Goodness-of-fit tests for the spatial spectral density

“…A Bootstrap bias correction technique is also considered, in order to correct the bias of parameter estimates. A better behaviour of the testing methods was observed when including this correction, instead of considering the modified Whittle estimations, such as the one proposed by Guyon (1982).…”

“…In order to obtain a ffiffiffiffi N p -consistent estimator of h, an unbiased version of the periodogram can be used in the Whittle log-likelihood expression (see Guyon 1982). The unbiased periodogram is obtained from (3), replacing the sample covariancesĈðuÞ by the unbiased sample covariances, namelyĈ u ðuÞ; with u T = (u 1 , u 2 )…”

“…Whittle (1954) points out the difference between stationary processes in the plane or in time and derives general estimation equations, which provide the so called Whittle estimates. Guyon (1982) and Dahlhaus and Künsch (1987) propose corrections of Whittle estimates in order to achieve consistency for higher dimensional settings. The estimation problem in this context has been deeply studied.…”

On the Whittle Estimator of the Parameter of Spectral Density of Random Noise in the Nonlinear Regression Model

“…The results obtained in this field form a well-developed theory that covers a broad spectrum of the mathematical models of random processes and fields (see, e.g., [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]). In [23], the authors combine the available results obtained in estimating the parameters of the linear regression model with the results of estimation of the parameters of spectral density of random noise by the Whittle method in the case of discrete time and strong dependence of the errors of observations.…”