Parametric estimation of P(X > Y) for normal distributions in the context of probabilistic environmental risk assessment

Estimating the risk, P(X > Y), in probabilistic environmental risk assessment of nanoparticles is a problem when confronted by potentially small risks and small sample sizes of the exposure concentration X and/or the effect concentration Y. This is illustrated in the motivating case study of aquatic risk assessment of nano-Ag. A non-parametric estimator based on data alone is not sufficient as it is limited by sample size. In this paper, we investigate the maximum gain possible when making strong parametric assumptions as opposed to making no parametric assumptions at all. We compare maximum likelihood and Bayesian estimators with the non-parametric estimator and study the influence of sample size and risk on the (interval) estimators via simulation. We found that the parametric estimators enable us to estimate and bound the risk for smaller sample sizes and small risks. Also, the Bayesian estimator outperforms the maximum likelihood estimators in terms of coverage and interval lengths and is, therefore, preferred in our motivating case study.