格子点上随机场的回归系数的估计问题

<正> §1.引言U.Grenander 研究了随机叙列的回归系数的估计问题,最近 M.Rosenblatt 研究了随机向量叙列的回归系数的估计问题.我们这桌案里研究格子点上随机场的回归系数的估计问题.前二作者所采用的方法是一样的,但是对于随机场而言若采用同一方法则有

ON THE ESTIMATION OF REGRESSION COEFFICIENTS OF A RANDOM FIELD WITH HOMOGENEOUS RESIDUAL

U.Grenander has investigated the problem of estimating the regression coefficients of a time series with stationary residual.Recently M.Rosenblatthas generalized to the case of vector valued time series with stationary resi-dual.In this paper we investigate the problem of estimating the regressioncoefficients of a random field defined on the positive lattice points in R_k.Byour method we can extend the asymptotic efficiency formula to more generaltype of spectral functions of the residual field.For simplicity we consideronly the case that there is only one regression coefficient.Let the randomfield{x_(n_1),…,n_k}n_j=0,1,2,…j=1,2,…,k with expectation cψ_(n_1)…n_k beexpressed asx_(n_1)…n_k=cψ_(n_1)…n_k+y_(n_1)…n_k, (1)where{ψ_(n_1)…n_k}is a known sequence,they are called the regression variablesand c an unknown constant,it is called the regression coefficient.The resi-dual field{y_(n_1)…n_k}is assumed to be a homogeneous random field with spectralfunction F(λ_1…λ_k).Hence the correlation function of the random field canbe written as(?)(2)We consider two kinds of estimates of c;one is the least square estimateC_Ls,the other is the best linear unbiassed estimate C_(BLS).Let(?)We first assume that as N_j→∞(j=1,…,k)(denoted by N→∞),Φ_N→∞.Some conditions of this type is required if we are to be able to estimate cconsistently.We also require that for any h=(h-1,…,h_k),(?)(3)If one of the ν's is a negative integer,we set ψ_(ν1)…νk=0,Let the limit(?)(4)exist for all(ν_1…ν_k).Then the condition(3)is equivalent to(?)for any(h_1…h_k),and the sequence{ψ_(ν1)…νk}is called homogeneous,thenR_(ν1)…νk has the following spectral representation:(?)(5)where the function ψ(λ)=ψ(λ_1…λ_k)is a distribution law on the k-dimensionalinterval-1/2,1/2;…;-1/2,1/2),ψ(λ)is called the spectral function of thehomogeneous sequence{ψ_(ν1…νk)}.In this paper are have proved the following:Theorem 3.Let{ψ_(n_1)…n_k}be a homogeneous sequence with sepectral function ψ(λ),then at every continuity interval I of ψ,we have(?)(6)Using this theorem and under furthor assumptions on the spectral functionF(λ)of the homogeneous residual we can deduce the following theorems,whichare generalizations of Grenander's~(1])results about the asymptotic efficiency ofthe least square estimate C_(LS)of c.Theorem 6.Let the regression variables ψ_(ν1)…νk of the random field x_(ν_1)…νkbe homogeneous,whose spectral function be ψ(λ)=ψ(λ_1…λ_k).Let the spec-tral function F(λ)of the homogeneous residual random field y_(ν1)…νk)can bedecomposed into the following form(?)(7)where f(λ)=f(λ_1…λ_k)is a positive continuous function in-1/2,1/2;…;-1/2,1/2],H is a singular measure satisfying the following condition:(?)(8)Then the asymptotic efficiency of the least square estimate C_(LS) is given by thefollowing expression:(?)Corollary 6.1.Under the conditions of Theorem 6,in order that theleast square estimate shall be asymptotically efficient whatever be the functionf(λ)as described in Theorem 6,it is necessary and sufficient that the wholemass of ψ must be concentrated at a single point.From our method,it seems impossible to extend Theorem 6 and its corol-lary to a class of residual with spectral function considerably general than thosewe have treated here.However we can prove the following theorem in whichthe conditions imposed on the spectral function of the residual is extended tosuch a level of generality as in U.Grenander's work on the equidistributedestimate of the mean value of a process6].Theorem 7.Let the regression variables{ψ_ν1…νk}be homogeneous withspectral function ψ(λ)=ψ(λ_1…λ_k).Suppose that the mass of ψ is concen-trated at a point P_0.Further let the spectral function F(λ)of the residualy_(ν1…νk) can be decomposed into the following form(?)where H is a singular measure satisfying the condition(8),and f(λ)=f(λ_1…λk)is a positive bounded function continuous at a neighborhood of P_0,then theleast square estimate is asymptotically efficient.