Some test statistics for the structural coefficients of the multivariate linear functional relationship model
The multivariate linear functional relationship (MLFR) model is written as follows: The p-dimensional observations \(\underset \tilde{} x_{ij}\) are assumed to be generated by \(\underset \tilde{} x_{ij}=\underset \tilde{} g_ i+{\underset \tilde{} \epsilon}_{ij}\), \(i=1,...,r\); \(j=1,...,n_ i\), where the \({\underset \tilde{} \epsilon}_{ij}\) are independent \(N_ p(\underset \tilde{} 0,\underset \tilde{} V)\) for \(\underset \tilde{} V\) positive definite, and \(\underset \tilde{} g_ 1,...,\underset \tilde{} g_ r\) are solutions of \(\underset \tilde{} B'\underset \tilde{} g=-\underset \tilde{} a\), with \(\underset \tilde{} B'\) of size \(k\times p\), and full rank k \((k<p)\). The hypothesis of interest is \(H_ 0:\underset \tilde{} a=\underset \tilde{} a_ 0\), \(\underset \tilde{} B=\underset \tilde{} B_ 0\), and \(\underset \tilde{} V\) unknown. A test statistic by \textit{A. P. Basu} [J. R. Stat. Soc., Ser. B 31, 65-71 (1969; Zbl 0176.485)] is considered, and since it depends on the unknown \(\underset \tilde{} V\), two alternative test procedures for \(H_ 0\) are derived. One test statistic is \(Q=r\underset \tilde{} t\bar {\;}'\underset \tilde{} S^{*-1}\underset \tilde{} t\bar {\;}\), where \(\underset \tilde{} t\bar {\;}\) is the average of the \(\underset \tilde{} t_ i=n_ i(\underset \tilde{} B_ 0'\underset \tilde{} B_ 0)^{-}(\underset \tilde{} a_ 0+\underset \tilde{} B_ 0'\underset \tilde{} x\bar {\;}_ i)\), and \(\underset \tilde{} S^*=\sum^{r}_{i=1}(\underset \tilde{} t_ i- \underset \tilde{} t\bar {\;})(\underset \tilde{} t_ i-\underset \tilde{} t\bar {\;})'\); its distribution under \(H_ 0\) is k \(F_{k,n-k}/(n-k).\) Another test statistic for \(H_ 0\) is \(Q_ c=n\underset \tilde{} c\bar {\;}'\underset \tilde{} S_ c^{*-1}\underset \tilde{} c\bar {\;}\) defined similarly for \(\underset \tilde{} c_{ij}=(\underset \tilde{} B_ 0'\underset \tilde{} B_ 0)^{- }(\underset \tilde{} a_ 0+\underset \tilde{} B_ 0'\underset \tilde{} x_{ij})\), having the same distribution as Q under \(H_ 0.\) A final section of the paper considers in detail the case where \(\underset \tilde{} V\) is known up to a constant of proportionality, that is, the hypothesis is modified by setting \(\underset \tilde{} V=\nu^ 2\underset \tilde{} V_ 0\) with \(\underset \tilde{} V_ 0\) known.
multivariate linear functional relationship
scientific article; zbMATH DE number 4022402
scientific article