  We study holonomic gradient decent for maximum likelihood estimation of
exponential-polynomial distribution, whose density is the exponential function
of a polynomial in the random variable. We first consider the case that the
support of the distribution is the set of positive reals. We show that the
maximum likelihood estimate (MLE) can be easily computed by the holonomic
gradient descent, even though the normalizing constant of this family does not
have a closed-form expression and discuss determination of the degree of the
polynomial based on the score test statistic. Then we present extensions to the
whole real line and to the bivariate distribution on the positive orthant.
