Distribution Fitting with the Quantile Function of Response Modeling Methodology (RMM)

Distribution fitting aims to provide the data analyst with a general platform for modeling random variation. The need for such a “general-purpose” platform arises when either the available sample size is too small, or a-priori knowledge about the source of the data too scarce to identify with any acceptable degree of confidence the true underlying distribution. In such circumstances, distribution fitting proposes to substitute the unknown distribution by a member of a multi-parameter family of distributions, like the Johnson or Pearson families. It is assumed that such (known) families are flexible enough to deliver good representation to unknown distributions, no matter how diversely-shaped they might be in practice. Response Modeling Methodology (RMM), developed in the early nineties of the previous century (Shore, 2005, and references therein), provides such a platform. Although originally developed as a general methodology for empirical modeling of systematic variation (variation in a response traceable to variation of predictor variables correlated with the response), the quantile function of the RMM error distribution has been shown to deliver good representation to a wide range of variously shaped distributions. Furthermore, RMM reduces to some well-known distributions, approximations and transformations for selected values of its parameters (Shore, (2004a), and (2005), Chapter 12).
In this chapter, we first outline the rationale for using an RMM quantile function as a general platform for distribution fitting (Section 15.1). In the following Section 15.2 we derive the quantile function of RMM’s original model and some variations. Estimation via maximum likelihood, percentile matching and moment matching are addressed