Unbiased estimation of P(Y &lt; X) in the normal case

Voinov, V.G.

doi:10.1007/bf01091435

Cited by 3 publications

(3 citation statements)

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“…In engineering, R = P ( X > Y ) is known as the reliability in stress–strength models. This is a well-known concept and has been studied extensively for the normal–normal model (Barbiero, 2011; Church & Harris, 1970; Downtown, 1973; Enis & Geisser, 1971; Govidarajulu, 1967; Nandi & Aich, 1996; Voinov, 1986; Weerahandi & Johnson, 1992) as well as for other distributions (Kundu & Gupta, 2006; Mokhlis, 2005; Nadar, Kızılaslan & Papadopoulos, 2014). None of these papers give sufficiently general theory for obtaining trustworthy interval estimates in the case study.…”

Section: Introductionmentioning

confidence: 99%

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

Jacobs

Bekker

Voet

et al. 2015

PeerJ

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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.

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Section: Introductionmentioning

confidence: 99%

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

Jacobs

Bekker

Voet

et al. 2015

PeerJ

View full text Add to dashboard Cite

show abstract

“…In engineering, R = P (X > Y ) is known as the reliability in stress-strength models. This is a well-known concept and has been studied extensively for the normal-normal model (Barbiero, 2011;Church and Harris, 1970;Downtown, 1973;Enis and Geisser, 1971;Govidarajulu, 1967;Nandi and Aich, 1996;Voinov, 1986;Weerahandi and Johnson, 1992) as well as for other distributions (Kundu and Gupta, 2006;Mokhlis, 2005;Nadar et al, 2014). None of these papers gives sufficiently general theory for obtaining trustworthy interval estimates in the case study.…”

Section: Introductionmentioning

confidence: 99%

Peer Review #1 of "Parametric estimation of P(X > Y) for normal distributions in the context of probabilistic environmental risk assessment (v0.1)"

2015

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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 nonparametric 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.

show abstract

“…In engineering, R = P (X > Y ) is known as the reliability in stressstrength models. This is a well-known concept and has been studied extensively for the normal-normal model (Barbiero, 2011;Church and Harris, 1970;Downtown, 1973;Enis and Geisser, 1971;Govidarajulu, 1967;Nandi and Aich, 1996;Voinov, 1986;Weerahandi and Johnson, 1992) as well as for other distributions (Kundu and Gupta, 2006;Mokhlis, 2005;Nadar et al, 2014). None of these papers gives sufficiently general theory for obtaining trust-worthy interval estimates in the case study.…”

Section: Resultsmentioning

confidence: 99%

Statistical modelling of variability and uncertainty in risk assessment of nanoparticles

Jacobs

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Engineered nanoparticles (ENPs) are used everywhere and have large technological and economic potential. Like all novel materials, however, ENPs have no history of safe use.Insight into risks of nanotechnology and the use of nanoparticles is an essential condition for the societal acceptance and safe use of nanotechnology.Risk assessment of ENPs has been hampered by lack of knowledge about ENPs, their environmental fate, toxicity, testing considerations, characterisation of nanoparticles and human and environmental exposures and routes. This lack of knowledge results in uncertainty in the risk assessment. Moreover, due to the novelty of nanotechnology, risk assessors are often confronted with small samples of data on which to perform a risk assessment. Dealing with this uncertainty and the small sample sizes are main challenges when it comes to risk assessment of ENPs. The objectives of this thesis are (i) to perform a transparent risk assessment of nanoparticles in the face of large uncertainty in such a way that it can guide future research to reduce the uncertainty and (ii) to evaluate empirical and parametric methods to estimate the risk probability in the case of small sample sizes.To address the first objective, I adapted an existing Integrated Probabilistic Risk Assessment (IPRA) method for use in nanoparticle risk assessment. In IPRA, statistical distributions and bootstrap methods are used to quantify uncertainty and variability in the risk assessment in a two-dimensional Monte Carlo algorithm. This method was applied in a human health (nanosilica in food) and an environmental (nanoT iO 2 in water) risk context. I showed that IPRA leads to a more transparent risk assessment and can direct further environmental and toxicological research to the areas in which it is most needed.For the second objective, I addressed the problem of small sample size of the critical effect concentration (CEC) in the estimation of R = P (E xpC > C EC ), where E xpC is the exposure concentration. First I assumed normality and investigated various parametric and non-parametric estimators. I found that, compared to the non-parametric estimators, the parametric estimators enable us to better estimate and bound the risk when sample sizes and/or small risks are small. Moreover, the Bayesian estimator out-ABSTRACT performed the maximum likelihood estimators in terms of coverage and interval lengths.Second, I relaxed the normality assumption for the tails of the exposure and effect distributions. I developed a mixture model to estimate the risk, R = P (E xpC > C EC ), with the assumption of a normal distribution for the bulk data and generalised Pareto distributions for the tails. A sensitivity analysis showed significant influence of the tail heaviness on the risk probability, R, especially for low risks.In conclusion, to really be able to focus the research into the risks of ENPs to the most needed areas, probabilistic methods as used and developed in this thesis need to be implemented on a larger scale. With these methods,...

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Unbiased estimation of P(Y < X) in the normal case

Cited by 3 publications

References 5 publications

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

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

Peer Review #1 of "Parametric estimation of P(X > Y) for normal distributions in the context of probabilistic environmental risk assessment (v0.1)"

Statistical modelling of variability and uncertainty in risk assessment of nanoparticles

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