A source density function

The delta function initial condition solution of the generalized Feller equation gives rise to a four‐parameter class of one‐sided probability source density functions. Because of this origin, each of these functions represents the instantaneous density of an autonomous Markov diffusion process. As the source location parameter approaches zero the density class to be investigated here reduces to the hyper‐Gamma class. Because of the immense flexibility in shape (as compared with the hyper‐Gamma class) the source density class offers itself to a wide variety of applications in physics and statistics, especially if the observations are far removed from zero. Relative to applications, the natural and paramount concern must be directed toward the parameter estimation problem. This paper provides the analytical foundation for a numerical approach to its solution. The log‐likelihood function for the source density class will be established. Based on it the maximum‐likelihood equations will be derived. For practical numerical reasons, direct maximization of the log‐likelihood function may be more successful than an attempt to solve the (three) maximum‐likelihood equations. Therefore, the analytical properties of the log‐likelihood function will be discussed, as they are essential in the implementation of any numerical optimization routine.