In this paper, we study generalized linear models under distribution uncertainty, where the random intercept has mean uncertainty and takes values in an unknown bounded interval. We develop a novel two-step method, named Firth-corrected Expectation-Maximization with Moving Block (FEMMB), to estimate the regression coefficients as well as the lower and upper bounds of the intercept. A Firth bias correction is incorporated into the EM algorithm to mitigate the incidental parameters problem, which is prominent in generalized linear models with many nuisance intercepts. Under mild regularity conditions, we establish the consistency and asymptotic normality of the estimated regression coefficients, and the consistency of the estimated intercept bounds. Simulation studies and a real-data application to an imbalanced credit card dataset demonstrate the effectiveness of the proposed method.

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