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Empirical Survival Jensen-Shannon Divergence as a Goodness-of-Fit Measure for Maximum Likelihood Estimation and Curve Fitting

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  • Mark Levene
  • Aleksejus Kononovicius

Abstract

The coefficient of determination, known as $R^2$, is commonly used as a goodness-of-fit criterion for fitting linear models. $R^2$ is somewhat controversial when fitting nonlinear models, although it may be generalised 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 (${\cal E}SJS$) as a metric which is bounded, and is a natural generalisation of the Jensen-Shannon divergence. We demonstrate, via a straightforward procedure making use of the ${\cal E}SJS$, 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.

Suggested Citation

  • Mark Levene & Aleksejus Kononovicius, 2018. "Empirical Survival Jensen-Shannon Divergence as a Goodness-of-Fit Measure for Maximum Likelihood Estimation and Curve Fitting," Papers 1809.11052, arXiv.org, revised Jun 2019.
  • Handle: RePEc:arx:papers:1809.11052
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    References listed on IDEAS

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    1. Trevor Fenner & Mark Levene & George Loizou, 2018. "A multiplicative process for generating the rank-order distribution of UK election results," Quality & Quantity: International Journal of Methodology, Springer, vol. 52(3), pages 1069-1079, May.
    2. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    3. Vuong, Quang H, 1989. "Likelihood Ratio Tests for Model Selection and Non-nested Hypotheses," Econometrica, Econometric Society, vol. 57(2), pages 307-333, March.
    4. Colin Cameron, A. & Windmeijer, Frank A. G., 1997. "An R-squared measure of goodness of fit for some common nonlinear regression models," Journal of Econometrics, Elsevier, vol. 77(2), pages 329-342, April.
    5. Aleksejus Kononovicius, 2017. "Empirical Analysis and Agent-Based Modeling of the Lithuanian Parliamentary Elections," Complexity, Hindawi, vol. 2017, pages 1-15, November.
    6. Nadarajah, Saralees & Kotz, Samuel, 2007. "On the q-type distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 377(2), pages 465-468.
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    Cited by:

    1. Aleksejus Kononovicius & Bronislovas Kaulakys, 2022. "$1/f$ noise from the sequence of nonoverlapping rectangular pulses," Papers 2210.11792, arXiv.org, revised Mar 2023.
    2. Levene, Mark & Fenner, Trevor, 2021. "A stochastic differential equation approach to the analysis of the 2017 and 2019 UK general election polls," International Journal of Forecasting, Elsevier, vol. 37(3), pages 1227-1234.

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