一类空间自回归模型的强相合性

Strong Consistency in Spatial Autoregression Model

(∧αn,∧βn)表示在空间自回归模型Zij=αZi-1,j+βZi,j-1-αβZi-1,j-1+iεj中参数(α,β)的Guass-Newtor估计。根据已知的结论:当α=β=1时,{n32(∧αn-α,∧βn-β)}收敛于二元正态随机向量分布即limn{(n32(∧αn-α,∧βn-β))′}DN2(0,Γ),其中Γ=d iag(2.2)。利用双参数强鞅收敛定理,可以证明:当r<32时,nr(∧αn-α,∧βn-β)→0.a.e。

Let （^∧αn,^∧βn）denote the Gauus -Newton estimator of the parameter （α ,β） in the autoregression model Zij=αZi-1j＋βZij-1-αβZi-1j-1 ＋εij. It is shown in an earlier paper that when α=β=1,{n^3/2（^∧α-α,^∧βn-β）} converges in distribution to a bivariate normal random randon vector：lim n {（n^3/2（^∧αn-α,^∧βn-β））′}D→N2（0,Г）,Г=diag（ 2.2 ） o A two -parameter strong martingale convergence theorem is employed here to prove that almost surely when n^r（^∧αn-α,^∧βn-β）→0- almost wurely when r〈3/2.