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The Distributions of the Peak to Average and Peak to Sum Ratios Under Exponentiality

In: Advances in Directional and Linear Statistics

Author

Listed:
  • Tomasz J. Kozubowski

    (University of Nevada, Department of Mathematics and Statistics, MS 084)

  • Anna K. Panorska
  • Fares Qeadan

Abstract

Let E 1, E 2, …, E N be independent and identically distributed exponential random variables, and let $$Y = \vee _{i = 1}^N$$ and S = ∑ i = 1 N E i be their maximum and sum, respectively. We review distributional properties of the peak to sum and peak to average ratios, R = Y ∕ S and $$\tilde{R} = Y/(S/N)$$ respectively, with deterministic N, and provide an extension to the case where N is itself a random variable, independent of the {E j }. Our results include explicit formulas for the relevant density and distribution functions, which apply to any distribution of N, as well as a particular example with geometrically distributed N. An example from climatology shows modeling potential of these models.

Suggested Citation

  • Tomasz J. Kozubowski & Anna K. Panorska & Fares Qeadan, 2011. "The Distributions of the Peak to Average and Peak to Sum Ratios Under Exponentiality," Springer Books, in: Martin T. Wells & Ashis SenGupta (ed.), Advances in Directional and Linear Statistics, chapter 0, pages 131-142, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7908-2628-9_9
    DOI: 10.1007/978-3-7908-2628-9_9
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