Upper Limit Lognormal Distribution for Drop Size Data

Abstract

The upper limit lognormal (ULLN) distribution is a three‐parameter family of curves that offers two significant features to models of size frequency data: they have finite maxima and their shape ranges from symmetric to leftor right‐skewed to bimodal. Three estimation procedures for ULLN parameters are examined: sequential maximum likelihood, systemic maximum likelihood and the method of percentiles. These algorithms are compared numerically using randomly generated ULLN data, and conclude that sequential iteration through the maximum likelihood equations should not be used; percentile estimators are inaccurate, and often infeasible; systemic maximum likelihood yields feasible, accurate estimates, converges rapidly, and can be initialized readily with percentile estimates when the latter are feasible. Four drop‐size data sets are fitted with ULLN distributions that pass the one‐sample Kolmogorov‐Smirnov test, give good visual fits to the empirical data, and corroborate crude results from earlier studies, indicating that the ULLN model may be quite satisfactory for modelling the distribution of droplets emitted from irrigation sprinklers.