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Intrinsic autoregressions and related models on the two-dimensional lattice. (English) Zbl 0671.62082

If \(X_ i\), \(i=(i_ 1,i_ 2)\), is a random process on the two- dimensional lattice it is said to be intrinsic of order d if all finite linear combinations \(\sum \lambda_ ix_ i\) are stationary provided \(\sum \lambda_ ip_ i=0\) for all polynomials of order d. The finite linear combination is then called an increment. If all increments have an absolutely continuous spectrum then \[ E(\sum \lambda_ jX_ j \sum \lambda_ k'X_ k)=(2\pi)^{-2}\int \{\sum \lambda_ j\exp (ijw)\sum \lambda_ k'ex\quad p(-ikw)\}f(w)dw \] where \(w=(w_ 1,w_ 2)\) lies in \((-\pi,\pi]^ 2\). Among stationary processes on the plane have been singled out the autoregressions for which \[ E(X_ i| \quad X_ j,j\neq i)=(1-\sum_{N}a_ k)E(X_ i)+\sum_{N}a_ kX_{i+k} \] and \(\sum_{N}\) is a sum over a symmetric neighbourhood of N and \(a_ k=a_{-k}\). Then \(f(w)=\sigma^ 2\{1-\sum_{N}a_ k\cos kw\}^{-1}.\)
After these preliminaries the author considers the generalization of autoregressions to intrinsic models. It is not appropriate to ask, in two dimensions, that a certain increment be an autoregression for, as the author shows, \(\| w\|^ 2f(w)\) is not then integrable, as it should be. The author calls a process, \(X_ i\), an intrinsic autoregression if \[ f(w)=\sigma^ 2/P(w)\quad and\quad P(w)=1-\sum a_ k\cos kw \geq c\| w\|^{2/d+2}. \] The order of \(X_ i\) is the smallest such d. He then says that, if V is a finite set in the plane, and \(i\in V\), then if \(X_ i-\sum \mu_ jX_ j\) is an increment it is the best intrinsic predictor if it minimizes the variance of all such expressions. Here also \(\mu_ i=0\) for \(i\in V.\)
He proves that for an intrinsic autoregression the expression \(\sum a_ kX_{i+k}\) is the best intrinsic predictor of \(X_ i\) and conversely, as well as some further theorems dealing with interpolation and the fact that \(E\{(X_{i+k}-X_ i)\}\) increases as \(\log \| k\|\) with k. Estimation is based on minimizing \[ -\int \log [\sum a_ k\{1-\cos kw\}/(1-2^{-1}\cos w_ 1-2^{-1}\cos w_ 2)]d\quad w+\sum a_ k{\hat \gamma}(k), \]
\[ {\hat \gamma}(k)=2^{-1}(n-| k_ 1|)^{-1}(m-| k_ 2|)^{-1}\sum (X_ i-X_{i+k}\quad)^ 2, \] which is derived as an approximation to the Gaussian likelihood. The paper concludes with an application to Landsat data.
Reviewer: E.J.Hannan

MSC:

62M07 Non-Markovian processes: hypothesis testing
60G10 Stationary stochastic processes
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M20 Inference from stochastic processes and prediction
62M99 Inference from stochastic processes
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