A Measure of Representativeness of a Sample for Inferential Purposes

Abstract

After defining the concept of representativeness of a random sample, the author proposes a measure of how much the observed sample represents its parent distribution. This measure is called Representativeness Index. The same measure, seen as a function of a sample and of a distribution, will be called Representativeness Function. For a given sample it provides the value of the index for the different distributions under examination, and for a given distribution it provides a measure of the representativeness of its possible samples. Such Representativeness Function can be used in an inferential framework just as the likelihood function, since it gives to any distribution the ''experimental support'' provided by the observed sample. This measure is distribution-free and it is shown to be a transformation of the well-known Cramér-von Mises statistic. By using the properties of that statistic, criteria for providing set estimators and tests of hypotheses are introduced. The utilization of the representativeness function in many standard statistical problems is outlined through examples. The quality of the inferential decisions can be assessed with the usual techniques (MSE, power function, coverage probabilities). The most interesting examples turn out to be those of situations that are ''non-regular'', as for instance the estimation of parameters involved in the support of the parent distribution, or less explored (model choice).