In healthcare studies, count data sets measured with covariates often exhibit heterogeneity and contain extreme values. To analyse such count data sets, we use a finite mixture of regression model framework and investigate a robust estimation approach, called the L 2 E [D.W. Scott, On fitting and adapting of density estimates, Comput. Sci. Stat. 30 (1998), pp. 124-133], to estimate the parameters. The L 2 E is based on an integrated L 2 distance between parametric conditional and true conditional mass functions. In addition to studying the theoretical properties of the L 2 E estimator, we compare the performance of L 2 E with the maximum likelihood (ML) estimator and a minimum Hellinger distance (MHD) estimator via Monte Carlo simulations for correctly specified and gross-error contaminated mixture of Poisson regression models. These show that the L 2 E is a viable robust alternative to the ML and MHD estimators. More importantly, we use the L 2 E to perform a comprehensive analysis of a Western Australia hospital inpatient obstetrical length of stay (LOS) (in days) data that contains extreme values. It is shown that the L 2 E provides a two-component Poisson mixture regression fit to the LOS data which is better than those based on the ML and MHD estimators. The L 2 E fit identifies admission type as a significant covariate that profiles the predominant subpopulation of normal-stayers as planned patients and the small subpopulation of long-stayers as emergency patients.