Empirical survival Jensen-Shannon divergence as a goodness-of-Fit measure for maximum likelihood estimation and curve fitting

Abstract

The coefficient of determination, known as R², is commonly used as a goodness-of-fit criterion for fitting linear models. R² is somewhat controversial when fitting nonlinear models, although it may be generalized on a case-by-case basis to deal with specific models such as the logistic model. Assume we are fitting a parametric distribution to a data set using, say, the maximum likelihood estimation method. A general approach to measure the goodness-of-fit of the fitted parameters, which is advocated herein, is to use a nonparametric measure for comparison between the empirical distribution, comprising the raw data, and the fitted model. In particular, for this purpose we put forward the Survival Jensen-Shannon divergence (SJS) and its empirical counterpart ($\mathcal{E}\mathit{SJS}$) as a metric which is bounded, and is a natural generalization of the Jensen-Shannon divergence. We demonstrate, via a straightforward procedure making use of the $\mathcal{E}\mathit{SJS},$ that it can be used as part of maximum likelihood estimation or curve fitting as a measure of goodness-of-fit, including the construction of a confidence interval for the fitted parametric distribution. Furthermore, we show the validity of the proposed method with simulated data, and three empirical data sets.

Keywords:

• Divergence measures
• Goodness-of-fit
• Maximum likelihood
• Curve fitting
• Survival Jensen-Shannon divergence