PARAMETER ESTIMATION OF SPATIAL AR MODEL

Consider a stable AR model of two parameter spatial series {Xt, t∈N2}, i. e. {Xtt∈N2} is homogeneous and satisfies the following difference equationXt-sum from n=s∈【v,p] to asXt-s=Wt （t∈N2）where {Wt, t∈N2} is a two parameter white noise and the notation【3, p] expresses the set of two dimentional lattice points {（k1, k2）: 0≤k1≤p1, 0≤k2≤p2 but （k1, k2）≠（0, 0）}, and furthermore the two-variable polynomial1-sum from n=（s1,s2）∈【0,p] a（s1,s2） Z1k1Z2s2≠0（|Z1|≤1,|Z2|≤1）.In this paper, under frirly general conditions （it is required that {Wt} Satisfies the conditions of two-parameter martingale difference, which is much weaker than supposing {Wt} to be i. i. d.）, the author obtains strong consistency and asymptotic normality of the Y-W （LS） estimate of the AR parameters {as} whenever n1n2→∞, where n1 and denote the horizontal and vertical sampling width respectively.

Consider a stable AR model of two parameter spatial series {X(t), t is-an-element-of N2}, 'i. e. {X(s), t is-an-element-of N2} homogeneous and satisfies the following difference equation [GRAPHICS] where [W(t), t is-an-element-of N2} is a two parameter white noise and the notation < 0, p] expresses the set of two dimentional lattice points {(k1, k2): 0 less-than-or-equal-to k1 less-than-or-equal-to p1, 0 less-than-or-equal-to k2 less-than-or-equal-to p2 but (k1, k2) not-equal (0, 0)], and furthermore the two-variable polynomial [GRAPHICS] In this paper, under frirly general conditions (it is required that {W(t)} satifies the conditions of two-parameter martingale difference, which is much weaker than supposing {W(t)} to be i. i. d.), the author obtains strong consistency and asymptotic normality of the Y-W (LS) estimate of the AR parameters {a(s)} whenever n1n2 --> infinity, where n1 and n2 denote the horizontal and vertical sampling width respectively.