Is there a generalization of Pillai trace and the Hotelling-Lawley trace?

Question

In the setting of multivariate multiple regression (vector regressor and regressand), the four major tests for the general hypothesis (Wilk's Lambda, Pillai-Bartlett, Hotelling-Lawley, and Roy's Largest Root) all depend on the eigenvalues of the matrix $H E^{-1}$, where $H$ and $E$ are the 'explained' and 'total' variation matrices.

I had noticed that the Pillai and Hotelling-Lawley statistics could both be expressed as $$\psi_{\kappa} = \mbox{Tr}\left(H\left[\kappa H + E\right]^{-1}\right),$$ for, respectively, $\kappa = 1, 0$. I am looking at an application where the distribution of this trace, defined for the population analogues of $H$ and $E$, is of interest for the $\kappa = 2$ case. (modulo errors in my work.) I am curious if there is some known unification of the sample statistics for general $\kappa$, or some other generalization that captures two or more of the four classical tests. I realize that for $\kappa$ not equal to $0$ or $1$, the numerator no longer looks like a Chi-square under the null, and so a central F approximation seems questionable, so maybe this is a dead end.

I am hoping that there has been some research on the distribution of $\psi_{\kappa}$ under the null (i.e. the true matrix of regression coefficients is all zero), and under the alternative. I am interested particularly in the $\kappa = 2$ case, but if there is work on the general $\kappa$ case, I could, of course, use that.

Answer 1

I imagine that productive generalizations would come out from observations that

1. some of these tests are norms of the vector ${\rm spec}[HE^{-1}]=\{\lambda_1, \ldots, \lambda_p\}$, so Hotelling-Lawley's trace is the $l_1$ norm, $\| \{\lambda_1, \ldots, \lambda_p\} \|_1$, and Roy's largest root is the $l_\infty$ norm, $\| \{\lambda_1, \ldots, \lambda_p\} \|_\infty$.
2. some of these tests may be a norm of the matrix $HE^{-1}$, e.g., Roy's largest root is the spectral, or $l_2$, norm $\| H E^{-1} \|_2$.
3. some of the tests may be of the generalized entropy form, e.g., Hotelling-Lawley's trace is GE(1), Roy's largest root is GE($\infty$), and Wilks' $\Lambda$ is GE(-1) on $\{1+\lambda_1, \ldots, 1+\lambda_p\}$, up to a monotone transformation each.

When other norms or other generalized entropy parameters are entertained, other statistics may be arrived at that might be meaningful. I doubt that any of them would produce your $\psi_2$, though.