Best linear unbiased quantile estimators for environmental standards

Recent research has sought to develop a statistically based approach to setting environmental standards, prompted by Barnett and O'Hagan (1997) whose recommendations for a statistically verifiable ideal standard (SVIS) were endorsed by the Royal Commission on Environmental Pollution (1998). Many current environmental standards are set without due consideration of uncertainty and variation and are often based on poorly defined principles. In this article we propose an SVIS set on a population quantile where inferences about that quantile are achieved through best linear unbiased quantile estimation (BLUQE). We concentrate on estimation using small samples from the normal distribution with both parameters unknown, as is often the case in environmental problems. We investigate the efficiency of this estimator in comparison with a quantile estimator based upon the common sample estimators of mean and standard deviation, (X) over bar and S, respectively. From extensive simulation, we tabulate 5 per cent and 1 per cent critical values for the 0.95 and 0.99 BLUQE and develop an approximate significance testing procedure, which we demonstrate using river water quality data. We consider the difference in power between this approximate test and a coefficient of variation-based approach appealing to properties of the non-central t distribution. Finally, we examine ranked set sampling and compare best linear unbiased quantile estimation based on ranked set samples with that based on ordinary random samples, demonstrating impressive efficiency gains. Copyright (C) 2002 John Wiley Sons, Ltd.