Summary: {\it T. M. Pukkila} and {\it C. R. Rao} [Pattern recognition based on scale invariant discriminant functions. Center for Multivariate Analysis, Tech. Rep. No.8609 (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 complex 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, complex compositional Gaussian of type-1 and type-2 distributions. Further, a new class of complex lognormal 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 for the estimation of the density of directional and compositional Gaussian data is indicated.