An analytical family of correlation-weighted elastic-net estimators for multicollinear and high-dimensional Conway–Maxwell–Poisson regression

{Monte Carlo simulation is used to show gains in prediction, variable selection and computing time for high-dimensional Conway–Maxwell–Poisson regression.} Penalized estimation for Conway–Maxwell–Poisson (COMP) regression has relied on ridge- and Liu-type estimators, with LASSO-type selection introduced only recently. We propose a new family of correlation-weighted elastic-net estimators for this model. The family weights the elastic-net penalty by the correlation structure of the design, the dispersion of the response and the signal strength, and sets its scales by closed-form rules instead of cross-validation. It comprises correlation-weighted (CWEN) and adaptive (ACWEN) forms of four analytical rules, nests maximum likelihood and the Elastic Net, and its adaptive class attains the oracle property under suitable rate conditions. In simulation the estimators outperform cross-validated competitors in prediction and support recovery in most designs at a fraction of their computing time, and four applications from ecology, health economics and a clinical trial illustrate the method.