Ibrahim Abdelrazeq | Searcharxiv

arXiv2025

How to verify that a given process is a Lévy-Driven Ornstein-Uhlenbeck Process

Assuming that a Lévy-Driven Ornstein-Uhlenbeck (or CAR(1)) processes is observed at discrete times $0$, $h$, $2h$,$\cdots$ $[T/h]h$. We introduce a step-by-step methodological approach on how a person would verify the model assumptions. The methodology involves estimating the model parameters and approximating the driving process. We demonstrate how to use the increments of the approximated driving process, along with the estimated parameters, to test the assumptions that the CAR(1) process is Lévy-driven. We then show how to test the hypothesis that the CAR(1) process belongs to a specified class of Lévy processes. The performance of the tests is illustrated through multiple simulations. Finally, we demonstrate how to apply the methodology step-by-step to a variety of economic and financial data examples.

stat.AP↗

arXiv2020

On one-sample Bayesian tests for the mean

This paper deals with a new Bayesian approach to the standard one-sample $z$- and $t$- tests. More specifically, let $x_1,\ldots,x_n$ be an independent random sample from a normal distribution with mean $μ$ and variance $σ^2$. The goal is to test the null hypothesis $\mathcal{H}_0: μ=μ_1$ against all possible alternatives. The approach is based on using the well-known formula of the Kullbak-Leibler divergence between two normal distributions (sampling and hypothesized distributions selected in an appropriate way). The change of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Eliciting the prior, checking for prior-data conflict and bias are also considered. Many theoretical properties of the procedure have been developed. Besides it's simplicity, and unlike the classical approach, the new approach possesses attractive and distinctive features such as giving evidence in favor of the null hypothesis. It also avoids several undesirable paradoxes, such as Lindley's paradox that may be encountered by some existing Bayesian methods. The use of the approach has been illustrated through several examples.

math.ST↗