Distance of a Mixture from its Parent Distribution

Mixtures of distributions frequently appear in the theory and applications of proba-bility and statistics. However, the mixture distribution seldom has an explicit form. Then, we may envisage either to keep the parent distribution or to get an approx-imation of the mixture one. In such cases it is important to evaluate the distances between mixture distribution and its parent, or approximation. Therefore, an impor-tant literature is concerned with bounds on the distance between a mixture and its parent distribution (see for instance [2] and [1]). In this work, using a new approach, we study the distance of a mixture from its parent distribution. We consider mixtures of the form g(x) = f(x,m)Π(dm), where Π is a probability distribution and f(.,m) are parent density functions with mean m, for m in some setM. We are interested in finding bounds for the distance between the two densities g and f(.,M), for some fixed M inM. Our result is based on the polynomial expansion of the mixed densities f(.,m). We deduce expansions of the mixture density g and we express the difference between g and a parent density func-tion f(.,M) in terms of orthogonal polynomials. This expansion yields an expression of the distance in L1 and L2-norm. It provides also bounds for the difference between the mixture distribution function and its parent. Some illustrations are given for nor-mal, Poisson and gamma mixtures. We also study approximations derived from the polynomial expansion of the mixture density. In the same way we adapt our bounds for the distance of a mixture to such approximations