QMPE: estimating Lognormal, Wald, and Weibull RT distributions with a parameter-dependent lower bound

We describe and test quantile maximum probability estimator (QMPE), an open-source ANSI Fortran 90 program for response time distribution estimation. ¹QMPE enables users to estimate parameters for the ex-Gaussian and Gumbel (1958) distributions, along with three "shifted" distributions (i.e., distributions with a parameter-dependent lower bound): the Lognormal, Wald, and Weibull distributions. Estimation can be performed using either the standard continuous maximum likelihood (CML) method or quantile maximum probability (QMP; Heathcote & Brown, in press). We review the properties of each distribution and the theoretical evidence showing that CML estimates fail for some cases with shifted distributions, whereas QMP estimates do not. In cases in which CML does not fail, a Monte Carlo investigation showed that QMP estimates were usually as good, and in some cases better, than CML estimates. However, the Monte Carlo study also uncovered problems that can occur with both CML and QMP estimates, particularly when samples are small and skew is low, highlighting the difficulties of estimating distributions with parameter-dependent lower bounds.