A probability density function (PDF) describes the likelihood of different outcomes for a continuous random variable.

In some situations only the statistical properties of such objects are desired: the three-dimensional probability density function. This article demonstrates that under special symmetries this ...

The problem considered here is the estimation of the probability density function f (x1, ⋯, xp) at a point z = (z1, ⋯, zp) where f is positive and continuous. An estimator is proposed and consistency ...

Building on the widely-used double-lognormal approach by Bahra (1997), this paper presents a multi-lognormal approach with restrictions to extract risk-neutral probability density functions (RNPs) for ...