A Distribution-free Interval Mathematical Analysis of Probability Density Functions

In many real problems, such as in physical, biological, socio-economic sciences, medicine and certain natural sciences, one is faced with probabilistic models. To completely specify these models one must know the form of either probability distribution or density function. In practice one does not know the exact forms, rather one has to approximate these functions from given data. A great number of researchers have investigated different aspects of this problem. In this paper we concentrate on two broad classes of density estimators, namely kernel and spline function based estimators. The approach adopted is interval analysis, oriented so that the end computations can be easily and economically performed on modern computers. In order to understand clearly and simply, the ideas involved are unified by using the Hilbert spaces.