Regularization in simplex regression models: closed-form estimators and Monte Carlo evidence

{Asymptotic mean squared error analysis and Monte Carlo simulation are used to evaluate Ridge and Liu regularized estimators for simplex regression models under constant and varying dispersion, demonstrating significant improvements over maximum likelihood estimation under multicollinearity.} We address multicollinearity in simplex regression models, which accommodate both linear and nonlinear parameterizations and are well suited for modelling continuous data on the unit interval, such as proportions and rates. Emphasis is placed on the linear specification due to its practical relevance. Multicollinearity, a pervasive issue in high-dimensional settings, arises from strong dependence among covariates, inflating estimator variances and leading to unstable estimates and distortions in coefficient interpretation. To mitigate these effects, we propose Ridge and Liu shrinkage estimators for simplex regression models under constant and varying dispersion. Closed-form expressions are obtained by truncating the Fisher scoring algorithm after a single iteration, yielding a first-order asymptotic approximation with computational cost equivalent to a single weighted least squares fit. The finite-sample performance of the proposed estimators is evaluated through extensive Monte Carlo simulations and an application to a socioeconomic dataset with severe structural multicollinearity, showing improved stability relative to the unpenalized maximum likelihood estimator.