FINITE MIXTURES OF NATURAL EXPONENTIAL-FAMILIES

Let mu be a positive measure concentrated on R+ generating a natural exponential family (NEF) F with quadratic variance function V(F)(m), m being the mean parameter of F. It is shown that v(dx) = (gamma + x)mu(dx)(gamma greater-than-or-equal-to 0) generates a NEF G whose variance function is of the form l(mBAR) square-root DELTA(mBAR) + c-DELTA(mBAR), where l(mBAR) is an affine function of mBAR, DELTA(mBAR) is a polynomial in mBAR (the mean of G) of degree 2, and c is a constant. The family G turns out to be a finite mixture of F and its length-biased family. We also examine the cases when F has cubic variance function and show that for suitable choices of gamma the family G has variance function of the form P(mBAR) + Q(mBAR) square-root DELTA(mBAR) where P, Q are polynomials in mBAR of degree less-than-or-equal-to 2 while DELTA is an affine function of mBAR. Finally we extend the idea to two dimensions by considering a bivariate Poisson and bivariate gamma mixture distribution.