Theory of Regression and the Sampling Theory of Multidimensional Normal Distributions

Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 202)


Let ξ p+11,...,ξ p+1n , p⩽1 np+2 be r.v.’s possessing the following properties: E p+1i ) exists for 1⩾in and
$$ E({\xi _{p + 1i}}) = {\beta _0} + {\beta _1}{x_{1i}} + ... + {\beta _p}{x_{pi.}} $$
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⩾jp, 1⩾in are given real numbers and the βi0⩾ip, as well as the u ij, 1⩾i, jn, real parameters. The βi satisfy -<∞βi<∞ 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 p+1i0,...,βp) or even by E p+1i0,...,βp, uij, 1⩾i,jn) but the abbreviated notation should cause no misunderstanding. Our task is to construct unbiased estimates for each βi0⩾ip. 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,...,βp, uij,1⩾i,jn that (1.1) holds and (uij)\( _{1n}^{1n} \) 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.


Covariance Matrix Discriminant Function Joint Distribution Conditional Distribution Unbiased Estimate 
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    The bj provided they exist, will in general also depend on the xq, xp. Since, however, we view these n-tuples here as given, we suppress this dependence.Google Scholar
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    The notation here is so chosen that Xj is a k-dimensional vector but xj is the real number \( \frac{1}{{{n_1}}}\sum\limits_{i = 1}^{{n_1}} {{x_{ji}}} \) and similarly for the yji.Google Scholar
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    The symbol (ui.k...1) stands for the (p-K+1)-dimensional r.v. (up+1.k...1,...,uk+1.k...1).Google Scholar
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    First found by V. Romanovskij, Bull. Acad. Sci. Leningrad (6) 20, 643–648 (1926) and K. Pearson, Proc. Roy. Soc. London Ser. A, 112,1–14 (1926).Google Scholar
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    It is not entirely correct to use the symbol K2 p+1 here, but this should cause no confusion.Google Scholar
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© Springer-Verlag Berlin · Heidelberg 1974

Authors and Affiliations

  1. 1.University of ViennaAustria

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