Best Uniform Approximation to Probability Densities in Econometrics

A new method of approximating the probability density function (pdf’s) of econometric estimators and test statistics is developed. It is shown that best uniform approximants to a general class of pdf’s exist in the form of rational functions. A procedure for extracting approximants is devised and is based on modifying multiple-point Pade approximants to the distribution. The new approximation technique is very general and should be widely applicable in mathematical statistics and econometrics. It has the advantage, unlike the Edgeworth and saddlepoint approximations, of readily incorporating extraneous information on the distribution, even qualitative information. The new procedure is applied to a simple simultaneous equation estimator and gives exceptionally accurate results even for tiny values of the concentration parameter.