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Review of median stable distributions and Schröder’s equation

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  • Bassett, Gib

Abstract

Median stable distributions are an extension of traditional (mean) stable distributions. The traditional definition of stability (in terms of sums of iid random variables) is recast as a condition on the sampling distribution of an estimator. For the traditional (mean) stable distribution, the sample mean’s (rescaled) sampling distribution is identical to the distribution of the iid data. Median stable distributions are defined similarly by replacing the sample mean with the sample median. Since the sampling distribution of the median is a functional its stable distribution is the solution to a functional equation. It turns out that this defining functional equation is an instance of a famous equation due to Schröder from 1870. The fame of the equation is due to the way it incorporates iteration of functions, a key feature of what many years later would become dynamic systems analysis. The current paper reviews median stable distributions in light of its connection to Schröder’s functional equation.

Suggested Citation

  • Bassett, Gib, 2019. "Review of median stable distributions and Schröder’s equation," Journal of Econometrics, Elsevier, vol. 213(1), pages 289-295.
    • Handle: RePEc:eee:econom:v:213:y:2019:i:1:p:289-295
    DOI: 10.1016/j.jeconom.2019.04.016
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    References listed on IDEAS

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    1. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
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    More about this item

    Keywords

    Median stable distributions; Schröder’s functional equation;

    JEL classification:

    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics

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