Probability Models on Scale Invariant Functions for a Complex Random Vector

Pukkila and Rao (1986) have considered- probability models to scale invariant discriminant functions for real random variables. These models are extended to complex random variables. Let x be a compelx random vector and G(x) be a nonsingular measure of size such that G(λx) = λG(x) for all real λ > 0- Explicit expressions are obtained for the density of complex angular Gaussian (CAG), Complex compositional Gaussian of type-1 and type-2 distributions. Further, a new class of complex lognormal (CLN) distributions is defined and some results on these distributions are established. Some problems of estimation of the parameters are discussed, and a non-parametric method similar to Pukkila and Rao (1986) for the estimation of the density of directional and compositional Gaussian data is indicated.