Integrating probability and non-probability sampling data in presence of nonignorable participation

Non-probability samples are widely adopted in surveys for cost and efficiency, yet the absence of inclusion probabilities challenges finite population inference. A typical solution is integrating them with a reference probability sample using auxiliary information, but existing methods usually assume ignorable selection given covariates. In practice, participation often directly depends on the study variable, making selection nonignorable and standard adjustments invalid. This paper presents a unified likelihood-based framework for estimating finite population means under nonignorable non-probability sampling. The approach integrates empirical likelihood, pseudo-likelihood, and parametric modelling of the participation mechanism, and covers two common survey scenarios: (1) a reference sample with both auxiliary and study variables; (2) a reference sample with auxiliary variables only. Within this framework, we propose three estimators for the finite population mean and establish their asymptotic normality under suitable regularity conditions for survey contexts. Simulations confirm that the proposed estimators outperform existing methods in robustness and reliability when the ignorability assumption is violated. An application to the ESPACOV survey demonstrates the practical utility of the methodology.