Theory of Regression and the Sampling Theory of Multidimensional Normal Distributions

Abstract

Let ξ _p+11,...,ξ _p+1n , p⩽1 n⩽p+2 be r.v.’s possessing the following properties: E(ξ _p+1i ) exists for 1⩾i⩾n and

$$ E({\xi _{p + 1i}}) = {\beta _0} + {\beta _1}{x_{1i}} + ... + {\beta _p}{x_{pi.}} $$

(1.1)

Further let the covariance matrix of (ξ _p+11,...,ξ _p+1n ) exist and be denoted by U = (u _ij )$_{1n}^{1n}$. Here, the x _ji, 1⩾j⩾p, 1⩾i⩾n are given real numbers and the β_i0⩾i⩾_p, as well as the u _ij, 1⩾i, j⩾n, real parameters. The β_i satisfy -<∞β_i<∞ and the u _ij need only satisfy the trivial restriction that U be positive semi-definite. To be more precise, we should denote the right side of (1.1) by E(ξ _p+1i;β₀,...,β_p) or even by E(ξ _p+1i;β₀,...,β_p, u_ij, 1⩾i,j⩾n) but the abbreviated notation should cause no misunderstanding. Our task is to construct unbiased estimates for each β_i0⩾i⩾_p. In order to bring this problem into the general framework of V.1, we let the sample space be given by $({R_n},{\mathfrak{B}_n})$ and the set of joint distributions of (ξ _p+11,...,ξ _p+1n ) be so restricted by the parameters β₀,...,β_p, u_ij,1⩾i,j⩾n that (1.1) holds and (u_ij)$_{1n}^{1n}$ n is positive semi-definite. To obtain the estimates we make use of Gauss’ method of least squares, which is closely connected with the MLP.