Generalized Helmert-Ledermann orthogonal matrices and rom simulation

Deterministic rectangular orthonormal matrices satisfying a hyper-plane constraint plays a central role in random orthogonal matrix (ROM) simulation. The multivariate skewness and kurtosis sampling properties are encrypted in a given orthonormal matrix. We consider a subclass of generalized Helmert-Ledermann (GHL) orthogonal matrices that have fixed last column, are generated by the Cayley transform, and satisfy the required hyper-plane constraint. The algebraic structure of GHL orthogonal matrices is determined. Simple and convenient skewness and kurtosis formulas are obtained. We exhibit an asymptotically growing range of variation for skewness and kurtosis, which points to an increased flexibility in ROM simulation.