  This study introduces a new approach to addressing positive and unlabeled
(PU) data through the double exponential tilting model (DETM). Traditional
methods often fall short because they only apply to selected completely at
random (SCAR) PU data, where the labeled positive and unlabeled positive data
are assumed to be from the same distribution. In contrast, our DETM's dual
structure effectively accommodates the more complex and underexplored selected
at random PU data, where the labeled and unlabeled positive data can be from
different distributions. We rigorously establish the theoretical foundations of
DETM, including identifiability, parameter estimation, and asymptotic
properties. Additionally, we move forward to statistical inference by
developing a goodness-of-fit test for the SCAR condition and constructing
confidence intervals for the proportion of positive instances in the target
domain. We leverage an approximated Bayes classifier for classification tasks,
demonstrating DETM's robust performance in prediction. Through theoretical
insights and practical applications, this study highlights DETM as a
comprehensive framework for addressing the challenges of PU data.
