Deriving the Fisher information matrix for a multinomial distribution with non-proportional odds in a baseline category logit model

This paper derives the Fisher Information Matrix (FIM) using Maximum Likelihood Estimation (MLE) for a multinomial distribution with four categories under a baseline category logit model incorporating non-proportional odds assumptions. The baseline category logit model is a type of generalized linear model used for categorical response variables, where one category is treated as a baseline, and the log-odds of the other categories relative to the baseline are modelled as linear functions of the covariates. In this study, we consider a multinomial distribution with four outcome categories, and assume the odds are not proportional across the covariates, allowing the relationship between covariates and outcomes to vary by category. We derive the log-likelihood function for this model, then compute the first and second derivatives with respect to the model parameters to obtain the score function and the observed information matrix. The Fisher Information Matrix is then derived as the expected value of the observed information matrix. This derivation is essential for understanding the precision of parameter estimates obtained via MLE, as the FIM provides a measure of the information available in the data about the parameters. Additionally, the FIM is crucial for constructing confidence intervals and hypothesis tests for the parameters. Our results have broad applications in various fields such as epidemiology, social sciences, and marketing, where multinomial response models with complex covariate structures are commonly used. The theoretical development is complemented by a practical example demonstrating the implementation of these methods using simulated.