On the evaluation of an unknown probability density function, the direct estimation of the entropy from independent observations of a continuous random variable, and the distribution-free entropy test of goodness-of-fit

Instead of a histogram construction for the evaluation of an unknown PDF, the use of the uniform distribution of coverages of order statistics is proposed. This estimation makes it possible to evaluate the differential entropy of a continuous random variable directly through the distances between neighboring ordered observations. On the basis of this evaluation, a new nonparametric test can be constructed.