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A Markov Chain Monte Carlo comparison of variance estimators for the sampling of particulate mixtures


Hao Cheng, Faculty of Applied Science, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands.


During the sampling of particulate mixtures, samples taken are analyzed for their mass concentration, which generally has non-zero sample-to-sample variance. Bias, variance, and mean squared error (MSE) of a number of variance estimators, derived by Geelhoed, were studied in this article. The Monte Carlo simulation was applied using an observable first-order Markov Chain with transition probabilities that served as a model for the sample drawing process. Because the bias and variance of a variance estimator could depend on the specific circumstances under which it is applied, Monte Carlo simulation was performed for a wide range of practically relevant scenarios. Using the ‘smallest mean squared error’ as a criterion, an adaptation of an estimator based on a first-order Taylor linearization of the sample concentration is the best. An estimator based on the Horvitz–Thompson estimator is not practically applicable because of the potentially high MSE for the cases studied. The results indicate that the Poisson estimator leads to a biased estimator for the variance of fundamental sampling error (up to 428% absolute value of relative bias) in case of low levels of grouping and segregation. The uncertainty of the results obtained by the simulations was also addressed and it was found that the results were not significantly affected. The potentials of a recently described other approach are discussed for extending the first-order Markov Chain described here to account also for higher levels of grouping and segregation. Copyright © 2013 John Wiley & Sons, Ltd.