Maximum likelihood character of distributions

In this paper, some distribution in the family of those with invariance under orthogonal transformations within ans-dimensional linear subspace are characterized by maximun likelihood criteria. Specially, the main result is: supposeP _v is a projection matrix of a givens-dimensional subspaceV, andx ₁, ...,x _n > are i.i.d. samples drawn from population with a pdff(x′P _v x), wheref(·) is a positive and continuously differentiable function. ThenP _v (M _n ) is the maximum likelihood estimator ofP _v iff $$f(x) = c_k exp(kx)(k > 0)$$ where \(M_n = \sum\limits_{i = 1}^n {x_i x'_i ,P_u (M_n ) = \sum\limits_{i = 1}^n {\hat \xi _i \hat \xi '_i ,\lambda _1 , \cdot \cdot \cdot ,\lambda _2 } } \) are the firsts largest eigenvalues of matrixM _n , and \(\hat \xi _1 , \cdot \cdot \cdot ,\hat \xi _2 \) , are their associated eigenvectors.