SUMMARY The best invariant estimate of the parametric density function in statistical models invariant under a transformation group is derived. The estimate is best with respect to a goodness-of-fit criterion based on an informa,tion measure. We are concerned with the estimation of a parametric density function p(y I 0) using data x. Let r(y Ix) be an estimate of p(y I 0) and consider the goodness-of-fit criterion based on an information measure of Kullback & Liebler (1951), the deviation of r(y I x) from p(y I 0) being J= p'(xI0)dx p(y 0) log {p(y I 0)/r(y Ix)}dy, where p' is the density function of the data x. An estimate that minimizes J and is invariant under a group of transformations is said to be best invariant. Here we generalize the result of Murray (1977), who derived the best invariant estimate of the multivariate normal density function. Suppose that a class of parametric density functions {p(y I 0): 0 E E), y E Y} is postulated