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Statistical inference for functional relationship between the specified and the remainder populations. (English) Zbl 1284.62140

Summary: This paper is concerned with discovering linear functional relationships among \(k\) \(p\)-variate populations with mean vectors \(\mu_i\), \(i=1,\ldots ,k\), and a common covariance matrix \(\Sigma\). We consider a linear functional relationship to be one in which each of the specified \(r\) mean vectors, for example, \(\mu_1, \ldots, \mu_r\) are expressed as linear functions of the remainder mean vectors \(\mu_{r+1}, \ldots, \mu_k\). This definition differs from the classical linear functional relationship, originally studied by T. W. Anderson [Ann. Math. Stat. 22, 327–351 (1951; Zbl 0043.13902)], Y. Fujikoshi [J. Multivariate Anal. 4, 327–340 (1974; Zbl 0287.62026)] and others, in that there are \(r\) linear relationships among \(k\) mean vectors without any specification of \(k\) populations. To derive our linear functional relationship, we first obtain a likelihood test statistic when the covariance matrix \(\Sigma\) is known. Second, the asymptotic distribution of the test statistic is studied in a high-dimensional framework. Its accuracy is examined by simulation.

MSC:

62F05 Asymptotic properties of parametric tests
62J05 Linear regression; mixed models
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References:

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