We discuss here a general approach to the calculation of the
asymptotic covariance of $M$-estimates for location parameters in statistical
group models when an invariant objective function is used. The calculation
reduces to standard tools in group representation theory and the calculation of
some constants. Only the constants depend upon the precise forms of the density
or of the objective function. If the group is sufficiently large this
represents a major simplification in the computation of the asymptotic
covariance.
¶ Following the approach of Chang and Tsai we define a regression
model for group models and derive the asymptotic distribution of estimates in
the regression model from the corresponding distribution theory for the
location model. The location model is not, in general, a subcase of the
regression model.
¶ We illustrate these techniques using Stiefel manifolds.The Stiefel
manifold $\mathcal{V}_{p,m}$ is the collection of $p\times m$ matrices
$\mathbf{X}$ which satisfy the condition $\mathbf{X}^T \mathbf{X} =
\mathbf{I}_m$ where $m \le p$. Under the assumption that $\mathbf{X}$ has a
distribution proportional to $\exp(Tr(\mathbf{F}^T \mathbf{X}))$ for some
$p\times m$ matrix $\mathbf{F}$ Downs (1972) gives approximations to maximum
likelihood estimation of $\\mathbf{F}$. In this paper, we consider a somewhat
differenct location problem: under the assumption that $\mathbf{X}$ has a
distribution of the form $f(Tr(\mathbf{\theta}^T_0 \mathbf{X})$ for some
$\mathbf{\theta}_0 \in \mathcal{V}_{p,m}$, we calculate the asymptotic
distribution of $M$-estimates which minimize an objective function of the form
$\sum_i \rho(Tr(\mathbf{\theta}^T \mathbf{X}_i))$. i i The assumptions on the
form of the density and the objective function can be relaxed to a more general
invariant form. In this case, the calculation of the asymptotic distribution of
$\hat{\mathbf{\theta}}$ reduces to the calculation of four constants and we
present consistent estimators for these constants.
¶ Prentice (1989)introduced a regression model for Stiefel
manifolds. In the Prentice model, $\mathbf{u}_1, \mathbf{u}_2, \ldots,
\mathbf{u}_ n \in \mathcal{V}_{p,m}$ are fixed, $\mathbf{V}_1, \mathbf{V}_2,
\ldots, \mathbf{V}_ n \in \mathcal{V}_{p,m}$ are independent random so that the
distribution of $\mathbf{V}_i$ depends only upon $Tr(\mathbf{V}^T_i
\mathbf{A}_2 \mathbf{u}_i \mathbf{A}^T_1$ for unknown $(\mathbf{A}_1,
\mathbf{A}_2) \in SO(m) \times SO(p)$ .We discuss here M-estimation of
$\mathcal{A}_1$ and $\mathcal{A}_2$ under general invariance conditions for
both the density and the objective function.
¶ Using a well-studied example on vector cardiograms we discuss the
physical interpretation of the invariance assumption as well as of the
parameters (A1 A2) in the Prentice
regression model. In particular, A1 represents a rotation of
the u’s to the V’s in a coordinate systemrelative
to the u’s and A2 represents a rotation of the
u’s to the V’s in a coordinate system fixed to the
external world.