A general model of random variation

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A statistical distribution of a random variable is uniquely represented by its normal-based quantile function. For a symmetrical distribution it is S-shaped (for negative kurtosis) and inverted S-shaped (otherwise). As skewness departs from zero, the quantile function gradually transforms into a monotone convex function (positive skewness) or concave function (otherwise). Recently, a new general modeling platform has been introduced, response modeling methodology, which delivers good representation to monotone convex relationships due to its unique "continuous monotone convexity" property. In this article, this property is exploited to model the normal-based quantile function, and explored using a set of 27 distributions.

Original languageEnglish
Pages (from-to)1819-1841
Number of pages23
JournalCommunications in Statistics - Theory and Methods
Issue number9
StatePublished - 3 May 2015


  • Monotone convexity
  • Quantile function
  • Random variation
  • Response modeling methodology


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