Flexible Statistical Models for Imprecise and Uncertain Data: A New Generalization Approach
Loading...
Authors
Saah, David Kofi
Issue Date
2025
Type
Dissertation
Language
en_US
Keywords
Alternative Title
Abstract
We propose a general framework for extending any probability distribution ν to an absolutely continuous distribution with density g(y) =ν(y + A)/|A|, where A is any bounded measurable set of size |A|. This generalized distribution allows the model to adapt to data characteristics that the original (or “base”) distribution ν may not fully capture. We show that this generalized distribution corresponds to that of X + U, where X ~ ν and U is uniformly distributed over A, with X and U independent. This convolution-based framework enables the construction of new probability distributions by introducing additive uniform noise to a known parent distribution. Motivated by practical considerations such asimprecise measurements, data contamination, and truncation, the framework offers a flexible and analytically tractable approach to modeling distributional uncertainty. The key idea - representing the observed variable as the sum of a latent signal and bounded noise - produces generalized distributions that preserve the interpretability of the parent model while
accommodating real-world imperfections. This dissertation develops the theoretical foundations of the proposed construction, examines its properties, and explores its potential applications. In particular, we apply the framework to construct a new class of Extended Laplace (EL) distributions, designed to model Laplace data affected by independent uniform errors. We derive the fundamental properties of this EL distribution, propose a robust likelihood-based estimation method, and validate its performance through simulation studies. Applications in finance illustrate the EL model’s effectiveness in handling real-world data with inherent uncertainties. In the multivariate setting, we also develop a bivariate Bear-Claw distribution, which arises when the proposed scheme is applied to a specific bivariate exponential distribution. Overall, the proposed framework contributes a mathematically grounded, coherent approach to extending classical distributions while preserving interpretability and computational feasibility. Its ability to model uncertainty via bounded noise provides a valuable tool for modern statistical analysis of imperfect or incomplete data.