Reduction functions for the variance function of one-parameter natural exponential family

We show that, when a random variable has a parametric distribution as a member of an infinitely divisible natural exponential family whose induced measure is absolutely continuous with respect to its basis measure, there exists a deterministic function, referred to as “reduction function”, such that the random variable transformed by this function is an unbiased estimator of the variance of the random variable. Our result can be used in estimating latent structure in high-dimensional data and in implementing iterative reweighted least squares for generalized linear models.