Differentiated logdensity approximants

Abstract

A moment-based density approximation technique whereby the derivative of the logarithm of a density approximant is expressed as a rational function is introduced in this paper. Guidelines for the selection of the polynomial orders of the numerator and denominator are proposed. The coefficients are then determined by solving a system of linear equations. The resulting density approximation, referred to as a differentiated logdensity approximant or $\mathrm{DLA}$, satisfies a differential equation whose explicit solution is provided. It is shown that a unique solution exists when a polynomial is utilized in lieu of a rational function. The proposed methodology is successfully applied to two test statistics and several distributions. It is also explained that the same moment-matching technique can yield density estimates on the basis of sample moments. An example involving a widely analyzed data set illustrates this approach.