Estimating Percentiles of Bacteriological Counts of Recreational Water Quality Using Tweedie Models

Abstract

There are general guidelines and standards for measuring the microbial quality of water to prevent the incidence of disease outbreaks. Many agencies have chosen the 95th percentile; one can assess the recreational water quality, depending if the percentile value exceeds the guideline value or not. It is well known that this kind of data do not display a normal distribution and several alternatives have been proposed and are in use for estimating the percentile. A review of existing methods is given, that includes non parametric estimators as Hazen, Blom, Tukey and Weibull. We also describe transformations such as logarithmic and Box–Cox, that generate near normal data, after obtaining the normal percentile the inverse transformation is applied to obtain estimators in the original scale. A new methodology is proposed, consisting in finding the Tweedie distribution that better fits the observed data; this family has nonnegative support and can have a discrete mass at zero, making it useful to model skewed data that are a mixture of zeros and positive values. It allows working with parametric models in the original scale. We performed a Monte Carlo simulation to compare the performance of all the percentiles described above. As a result we noted that the percentile calculated from Tweedie distribution has lower mean square error than the others, which makes it the more precise estimator. All these techniques were applied to four data sets and, in all cases the Tweedie estimator was closer to the observed values than non parametric and anti transformed estimators.