Level sets and minimum volume sets of probability density functions

Abstract

Summarizing the whole support of a random variable into minimum volume sets of its probability density function is studied in the paper. We prove that the level sets of a probability density function correspond to minimum volume sets and also determine the conditions for which the inverse proposition is verified. The distribution function of the level cuts of a density function is also introduced. It provides a different visualization of the distribution of the probability for the random variable in question. It is also very useful to prove the above proposition. The volume λ of the minimum volume sets varies according to its probability α: smaller volume implies lower probability and vice versa and larger volume implies larger probability and vice versa. In this context, 1−α is the error of an erroneously classification of a new observation inside of the minimum volume set or corresponding level set. To decide the volume and/or the error of the level set that will serve to summarize the support of the random variable, a α−λ plot is defined. We also study the relation of the minimum volume set approach with random set theory when α is a random variable and extend the most specific probability–possibility transformation proposed in [System Theory, Knowledge Engineering and Problem Solving, in: Fuzzy Logic: State of the Art, vol. 12, Kluwer, 1993, pp. 103–112] to continuous piece-wise strictly monotone probability density functions.