Maximum likelihood character of distributions

In this paper, some distribution in the family of those with invariance under orthogonal transformations within an s-dimensional linear subspace are characterized by maximun likelihood criteria. Specially, the main result is: suppose P_v is a projection matrix of a given s-dimensional subspace V, and x₁, ..., x_n> are i.i.d. samples drawn from population with a pdf f(x′P_vx), where f(·) is a positive and continuously differentiable function. Then P_v(M_n) is the maximum likelihood estimator of P_v iff {Mathematical expression} where {Mathematical expression} are the first s largest eigenvalues of matrix M_n, and {Mathematical expression}, are their associated eigenvectors.