Abstract: To model categorical responses, multinomial logistic regressions with different links and parameter restrictions have widely been adopted based on the relationships among different categories. In this paper, a unified Poisson subsampling method is proposed to approximate efficiently the maximum likelihood estimator for regression parameters when big data are encountered. The asymptotic normality of the estimator generated from the Poisson subsample is established. Based on the derived asymptotic variance, optimal subsampling probabilities are given according to the A-optimality criterion. To mitigate the burden on the calculation of optimal subsampling probabilities, a random projection based procedure is applied. For practical implementation, some robustness issues including model misspecification and full data with possible outliers are further discussed with theoretical backups. The advantages of the proposed methods are illustrated through numerical studies on both simulated and real datasets.

Key words and phrases: Categorical data, Johnson–Lindenstrauss transform, Poisson subsampling, randomized Hadamard transform.