## 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

### Citations:

Zbl 0043.13902; Zbl 0287.62026
Full Text:

### References:

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