An evaluation of the methods of fractiles, moments and maximum likelihood for estimating parameters when sampling air quality data from a stationary lognormal distribution

Abstract

Air pollution control officials often make the simplifying assumption that air pollutant concentrations are independent samples from a stationary probability distribution. If the parent distribution really is stationary and correctly chosen, maximum likelihood estimation almost always will provide the best possible estimate of its parameters. However, the air pollution literature makes little if any mention of this fact and often suggests using the method of moments or the method of fractiles to estimate the parameters of an assumed distribution, and using the results for computing design values to determine the control level required to meet an air quality standard. No estimate is made in the air pollution literature of the magnitude of the difference produced by these different methods. This paper investigates the effectiveness of three different approaches for estimating parameters using a lognormal distribution as an example: (a) method of fractiles; (b) method of moments and (c) method of maximum likelihood. The error associated with each approach for computing emission controls is determined by sampling from a true stationary lognormal distribution using computer simulation. These results then are compared with a fourth approach, direct empirical linear rollback, in which no model is used and design values are calculated using raw observations. The latter approach often is used in practical situations by air pollution control personnel. In 100 simulated years at a site experiencing the same lognormally distributed air pollution in the precontrol state, the correct control level was 50%. The following control levels were calculated: Empirical rollback, 22–82%; Method of fractiles, 32–64%; Method of moments, 41–59% and Method of maximum likelihood, 46–54%, with most years very close to the true value of 50%. Thus, the maximum likelihood approach effectively reduces the variance by ‘filtering out’ the effect of random phenomena occurring during the year and would be the method of choice if the observations are indeed distributed as they are assumed.