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High-dimensional multinomial regression models are very useful in practicebut receive less research attention than logistic regression models, especiallyfrom the perspective of statistical inference. In this work, we analyze theestimation and prediction error of the contrast-based $\ell_1$-penalizedmultinomial regression model and extend the debiasing method to the multinomialcase, which provides a valid confidence interval for each coefficient and$p$-value of the individual hypothesis test. We apply the debiasing method toidentify some important predictors in the progression into dementia ofdifferent subtypes. Results of intensive simulations show the superiority ofthe debiasing method compared to some other inference methods.

$\ell_1$-penalized Multinomial Regression: Estimation, inference, and  prediction, with an application to risk factor identification for different  dementia subtypes