Novi Inverardi

Dec 4, 2024 | mdpi.com | Aldo Tagliani |Novi Inverardi |Pier Luigi

1. Introduction and Some Useful ToolsThe random variable X follows the lognormal distribution with parameters μ ∈ R and σ 2 > 0 , say X ∼ L N ( μ , σ 2 ) , if X is absolutely continuous with values in R + and has density function L N ( x ) = 1 2 π σ 2 x e − 1 2 σ 2 ( ln ( x ) − μ ) 2 , x ≥ 0 , (1) with integer moments μ k = e k μ + 1 2 k 2 σ 2 for k = 1 , 2 , ⋯ .

Mar 6, 2024 | mdpi.com | Aldo Tagliani |Novi Inverardi |Pier Luigi

1. IntroductionFor an absolutely continuous random variable X with the entire real axis as support and the associated Hamburger moment problem determinate, [1]—Thm.1 and [2] proved that the MaxEnt approximation converges in entropy to the unknown density f of X. Well-known necessary and sufficient conditions for the determinacy of the moment problem, as well as their geometric implications, were used to prove the above convergence result.

Jan 30, 2024 | mdpi.com | Jordan M. Stoyanov |Aldo Tagliani |Novi Inverardi |Pier Luigi

1. IntroductionWhen studying probability distributions, one of the challenging questions we arrive at comes from the classical moment problem. The question is whether or not a probability distribution is uniquely determined by the sequence of all moments, assuming they are finite. The answer can be given for the distribution itself; equivalently, for the associated random variable X; its distribution function F; the density f=F′; or the bounded positive measure μ=μF induced by F.

Dec 30, 2023 | mdpi.com | Aldo Tagliani |Novi Inverardi |Pier Luigi

1. IntroductionIn statistical estimation, one often wants to guess an unknown probability distribution F, given certain observations based on it. There are generally infinitely many distributions consistent with the available data, and the question of which of these to select is an important one in many fields. The notion of entropy has been proposed as a remarkable tool for performing this choice.

Sep 11, 2023 | mdpi.com | Pier Luigi |Aldo Tagliani |Jordan M. Stoyanov |Novi Inverardi

1.4. About the Novelties in Our ApproachCrucial in our approach is to exploit the following:The geometric interpretation of the indeterminacy conditions as developed by Merkes-Wetzel [21]. Properties of the eigenvalues of perturbed symmetric matrices in the spirit of Golub-Van Loan [22] and Wilkinson [23]. Both these are among the novelties in our exposition.