Distributions on an Interval as a Scale-Invariant Combination of Scale-Invariant Functions: Theoretical Explanation of Empirical Marchenko-Pastur-Type Distributions

Authors

Vladik Kreinovich, The University of Texas at El PasoFollow
Kevin Alvarez, The University of Texas at El PasoFollow
Chon Van Le, International University of Ho-Chi-Minh CityFollow

Publication Date

1-1-2021

Comments

Technical Report: UTEP-CS-21-04

Abstract

In many practical situations, we know the lower and upper bounds L and U on possible values of a quantity x. In such situations, the probability distribution of this quantity is also located on the corresponding interval [L, U]. In many such cases, the empirical probability distribution has the form d(x) = const * (x − L)α− * (U − x)α+ * xα. In the particular case α− = α+ = 0.5 and α = −1, we get the Marchenko-Pastur distribution that describes the distribution of the eigenvalues of a random matrix. However, in some cases, the empirical distribution corresponds to different values of α−, α+, and α. In this paper, we show that by using the general idea of scale-invariance, we can provide a theoretical explanation for the ubiquity of such Marchenko-Pastur-type distributions.