@article{zhao_frey_2006, title={Uncertainty for data with non-detects: Air toxic emissions from combustion}, volume={12}, ISSN={["1080-7039"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-33750414559&partnerID=MN8TOARS}, DOI={10.1080/10807030600977178}, abstractNote={ABSTRACT Air toxic emission factor datasets often contain one or more points below a single or multiple detection limits and such datasets are referred to as “censored.” Conventional methods used to deal with censored datasets include removing non-detects, replacing the censored points with zero, half of the detection limit, or the detection limit. However, the estimated means of the censored dataset by conventional methods are usually biased. Maximum likelihood estimation (MLE) and bootstrap simulation have been demonstrated as a statistically robust method to quantify variability and uncertainty of censored datasets and can provide asymptotically unbiased mean estimates. The MLE/bootstrap method is applied to 16 cases of censored air toxic emission factors, including benzene, formaldehyde, benzo(a)pyrene, mercury, arsenic, cadmium, total chromium, chromium VI and lead from coal, fuel oil, and/or wood waste external combustion sources. The proportion of censored values in the emission factor data ranges from 4 to 80%. Key factors that influence the estimated uncertainty in the mean of censored data are sample size and inter-unit variability. The largest range of uncertainty in the mean was obtained for the external coal combustion benzene emission factor, with 95 confidence interval of the mean equal to minus 93 to plus 411%.}, number={6}, journal={HUMAN AND ECOLOGICAL RISK ASSESSMENT}, author={Zhao, Yuchao and Frey, H. Christopher}, year={2006}, month={Dec}, pages={1171–1191} }
 @article{zhao_frey_2004, title={Development of probabilistic emission inventories of air toxics for Jacksonville, Florida}, volume={54}, ISSN={["2162-2906"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-8444251311&partnerID=MN8TOARS}, DOI={10.1080/10473289.2004.10471002}, abstractNote={Abstract Probabilistic emission inventories were developed for 1,3-butadiene, mercury (Hg), arsenic (As), benzene, formaldehyde, and lead for Jacksonville, FL. To quantify inter-unit variability in empirical emission factor data, the Maximum Likelihood Estimation (MLE) method or the Method of Matching Moments was used to fit parametric distributions. For data sets that contain nondetected measurements, a method based upon MLE was used for parameter estimation. To quantify the uncertainty in urban air toxic emission factors, parametric bootstrap simulation and empirical bootstrap simulation were applied to uncensored and censored data, respectively. The probabilistic emission inventories were developed based on the product of the uncertainties in the emission factors and in the activity factors. The uncertainties in the urban air toxics emission inventories range from as small as –25 to +30% for Hg to as large as –83 to +243% for As. The key sources of uncertainty in the emission inventory for each toxic are identified based upon sensitivity analysis. Typically, uncertainty in the inventory of a given pollutant can be attributed primarily to a small number of source categories. Priorities for improving the inventories and for refining the probabilistic analysis are discussed.}, number={11}, journal={JOURNAL OF THE AIR & WASTE MANAGEMENT ASSOCIATION}, author={Zhao, YC and Frey, HC}, year={2004}, month={Nov}, pages={1405–1421} }
 @article{frey_zhao_2004, title={Quantification of variability and uncertainty for air toxic emission inventories with censored emission factor data}, volume={38}, ISSN={["1520-5851"]}, url={http://www.scopus.com/inward/record.url?eid=2-s2.0-8544262958&partnerID=MN8TOARS}, DOI={10.1021/es035096m}, abstractNote={Probabilistic emission inventories were developed for urban air toxic emissions of benzene, formaldehyde, chromium, and arsenic for the example of Houston. Variability and uncertainty in emission factors were quantified for 71-97% of total emissions, depending upon the pollutant and data availability. Parametric distributions for interunit variability were fit using maximum likelihood estimation (MLE), and uncertainty in mean emission factors was estimated using parametric bootstrap simulation. For data sets containing one or more nondetected values, empirical bootstrap simulation was used to randomly sample detection limits for nondetected values and observations for sample values, and parametric distributions for variability were fit using MLE estimators for censored data. The goodness-of-fit for censored data was evaluated by comparison of cumulative distributions of bootstrap confidence intervals and empirical data. The emission inventory 95% uncertainty ranges are as small as -25% to +42% for chromium to as large as -75% to +224% for arsenic with correlated surrogates. Uncertainty was dominated by only a few source categories. Recommendations are made for future improvements to the analysis.}, number={22}, journal={ENVIRONMENTAL SCIENCE & TECHNOLOGY}, author={Frey, HC and Zhao, YC}, year={2004}, month={Nov}, pages={6094–6100} }