Statistical inference with non-probability survey samples is an emerging topic in survey sampling and official statistics and has gained increased attention from researchers and practitioners in the field. Much of the existing literature, however, assumes that the participation mechanism for non-probability samples is ignorable. In this paper, we develop a pseudo-likelihood approach to estimate participation probabilities for nonignorable non-probability samples when auxiliary information is available from an existing reference probability sample. We further construct three estimators for the finite population mean using regression-based prediction, inverse probability weighting (IPW), and augmented IPW estimators, and study their asymptotic properties. Variance estimation for the proposed methods is considered within the same framework. The efficiency of our proposed methods is demonstrated through simulation studies and a real data analysis using the ESPACOV survey on the effects of the COVID-19 pandemic in Spain.

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