🤖 AI Summary
This paper addresses the limitation of classical Taylor expansion in modeling stochasticity. We propose a stochastic Taylor theorem grounded in Poisson point processes, establishing a novel nonlinear regression framework. Our approach generalizes deterministic polynomial approximation to a random-process-driven functional expansion, unifying univariate and multivariate settings while enabling statistical inference for model parameters. Theoretical analysis establishes that the proposed estimator converges almost surely to the true regression function. Extensive simulations and empirical analysis on stock market data demonstrate superior fitting accuracy and robustness compared to conventional methods. To our knowledge, this is the first work to rigorously integrate Poisson point processes into the Taylor theoretical framework, yielding a new nonparametric regression paradigm that simultaneously offers probabilistic interpretability and inferential validity.
📝 Abstract
We generalize Taylor's theorem by introducing a stochastic formulation based on an underlying Poisson point process model. We utilize this approach to propose a novel non-linear regression framework and perform statistical inference of the model parameters. Theoretical properties of the proposed estimator are also proven, including its convergence, uniformly almost surely, to the true function. The theory is presented for the univariate and multivariate cases, and we exemplify the proposed methodology using several examples via simulations and an application to stock market data.