A New Approach to Small Sample Theory

It is argued that an integral part of the process by which the results of small sample theory can be transmitted to the applied econometrician will in the future be played by reliable direct approximation to the sampling and posterior distributions that are of interest in the precise setting of the model and the data set with which the investigator is working. The purpose of this paper is to introduce a new technique of approximating distributions which is sufficiently general to be widely used and is developed in very general terms and should be widely applicable in mathematical statistics and econometrics. It has the advantage, unlike existing methods that are based on asymptotic expansions, of readily incorporating extraneous information on the distribution; even qualitative or soft quantitative information, such as that based on Monte Carlo experiments. It is shown that best uniform approximants in the form of rational functions exist for a general class of probability density functions. Characterization, uniqueness and convergence theorems for these approximations are given. An operational procedure for extracting rational approximants with good global behavior is devised and is based on modifying multiple-point Padé approximants which will typically utilize purely local information about the behavior of the body and the tails of the distribution. The new procedure is applied to a simple simultaneous equation estimator and gives exceptionally accurate results even for tiny values of the concentration parameter.