Predictive inference for discrete-valued time series

2025-07-21

AI Summary

Constructing prediction intervals with controllable coverage probability for discrete-valued time series remains challenging due to inherent distributional discreteness and model uncertainty. Method: This paper proposes a novel probabilistic forecasting paradigm based on pre-specified target sets—estimating the probability that future observations fall within a user-defined set, rather than constructing conventional intervals. We develop asymptotic theory robust to model misspecification and design an adaptive bootstrap procedure to estimate the predictive probability distribution. The framework integrates integer-valued autoregressive (INAR) and INARCH models with conditional maximum likelihood estimation, accommodating both parametric and nonparametric settings, and systematically compares multiple bootstrap strategies. Contribution/Results: We establish theoretical consistency of the estimator and demonstrate, via simulations, high finite-sample coverage accuracy and compact predictive sets. Empirical analysis on real data further confirms the method’s effectiveness and practical utility.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks

Abstract

For discrete-valued time series, predictive inference cannot be implemented through the construction of prediction intervals to some predetermined coverage level, as this is the case for real-valued time series. To address this problem, we propose to reverse the construction principle by considering preselected sets of interest and estimating the probability that a future observation of the process falls into these sets. The accuracy of the prediction is then evaluated by quantifying the uncertainty associated with estimation of these predictive probabilities. We consider parametric and non-parametric approaches and derive asymptotic theory for the estimators involved. Suitable bootstrap approaches to evaluate the distribution of the estimators considered also are introduced. They have the advantage to imitate the distributions of interest under different possible settings, including the practical important case where uncertainty holds true about the correctness of a parametric model used for prediction. Theoretical justification of the bootstrap is given, which also requires investigation of asymptotic properties of parameter estimators under model misspecification. We elaborate on bootstrap implementations under different scenarios and focus on parametric prediction using INAR and INARCH models and (conditional) maximum likelihood estimators. Simulations investigate the finite sample performance of the predictive method developed and applications to real life data sets are presented.

Problem

• Predictive inference for discrete-valued time series
• Estimating probabilities for preselected future observation sets
• Evaluating prediction accuracy with uncertainty quantification

Innovation

• Reversed construction principle for discrete time series
• Parametric and non-parametric predictive probability estimation
• Bootstrap methods for uncertainty quantification