Probability density function is a statistical expression defining the likelihood of a series of outcomes for a continuous variable, such as a stock or ETF return.

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 ...

D. O. Loftsgaarden, C. P. Quesenberry, A Nonparametric Estimate of a Multivariate Density Function, The Annals of Mathematical Statistics, Vol. 36, No. 3 (Jun., 1965 ...

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 ...