Reply to Speckman and Rouder: A theoretical basis for QML

Abstract

Speckman and Rouder (2004) point out that Heathcote, Brown, and Mewhort (2002) did not prove that their quantile-based estimator (quantile maximum likelihood, or QML) approximates likelihood, but note that it outperformed an exact maximum likelihood method based on order statistics for small samples (n = 20) from the exGaussian distribution. When QML uses all order statistics as quantile estimates, it is equivalent to maximum spacings product (MSP) estimation (Cheng & Amin, 1983; Ranneby, 1984), which can be derived from information theory as a measure of model fit (Ekstrom, 2001). MSP has been shown to have asymptotic estimation properties that closely parallel conventional maximum likelihood (CML) for regular problems, and better performance in irregular problems, such as estimation of heavy-tailed distributions, mixtures of continuous distributions, and distributions with a minimum that must be estimated (shift distributions). CML can produce inconsistent estimates under these conditions, in particular for shift distributions commonly used in response time research, such as the lognormal, gamma, and Weibull, whereas MSP maintains efficiency or is hyperefficient.1 MSP is apparently obscure; neither we nor the reviewers of our earlier work were aware of MSP or its link to QML. We became aware of MSP in research for a paper on estimation software for shift distributions (Heathcote, Brown, & Cousineau, 2004). QML generalizes MSP by using linear combinations of order statistics (i.e., empirical quantile estimates). Titterington (1985) suggested a similar extension of MSP that uses averages of adjacent order statistics and is equivalent to the QML1 method that Heathcote et al. (2002) found to provide the best performance among the special cases of QML they examined. We conclude that QML1 is a viable alternative to MSP and that researchers can take advantage of the superior estimation properties of QML1 assured of consistency even in irregular cases. More work is needed to derive results for the efficiency of QML1, and for the consistency, efficiency, and asymptotic distribution of QML estimates based on alternative sets of quantiles (see Brown & Heathcote, 2003, for results related to asymptotic distribution).2 However, we agree with Speckman and Rouder (2004) that QML does not have an exact theoretical basis in likelihood (although it often approximates likelihood) and might better be named quantile maximum probability product (QMP) estimation. [Sidebar] (Manuscript received July 14, 2003; revision accepted for publication September 10, 2003.) NOTES 1. For example, Cheng and Amin (1983) showed that for the Gamma and Weibull distributions with shape parameter