Abstract
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.
Highlights
Like all novel materials, engineered nanoparticle (ENPs) have no history of safe use
We found that the maximum likelihood estimator (MLE) and the quasi maximum likelihood estimator (QMLE) had a similar pattern for both the bootstrap and noncentral t intervals, with the QMLE consistently having better coverage
In this paper we studied the problem of estimating the risk for the case of small sample size for effect concentrations and small R values
Summary
Like all novel materials, engineered nanoparticle (ENPs) have no history of safe use. This is the same as assuming normal distributions on the log-transformed exposure and effect concentrations This normal–normal model was developed in some detail by Aldenberg, Jaworska & Traas (2002) and allows an analytic expression for the risk when parameter values are known. R values and had some difficulty with the small sample sizes For these cases we calculated percentile confidence intervals. For the MLE-like estimators, we calculated confidence intervals based on the noncentral t distribution (Reiser & Guttman, 1986). In this method, the sum of the two variances (s2x and s2y) are approximated with a chi-squared distribution. Only consider the estimator based on Laplace’s Law of Succession
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