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The Discrete Weibull Distribution: An Alternative for Correlated Counts with Confirmation for Microbial Counts in Water

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  • James D. Englehardt
  • Ruochen Li

Abstract

Distributions of pathogen counts in treated water over time are highly skewed, power‐law‐like, and discrete. Over long periods of record, a long tail is observed, which can strongly determine the long‐term mean pathogen count and associated health effects. Such distributions have been modeled with the Poisson lognormal (PLN) computed (not closed‐form) distribution, and a new discrete growth distribution (DGD), also computed, recently proposed and demonstrated for microbial counts in water (Risk Analysis 29, 841–856). In this article, an error in the original theoretical development of the DGD is pointed out, and the approach is shown to support the closed‐form discrete Weibull (DW). Furthermore, an information‐theoretic derivation of the DGD is presented, explaining the fit shown for it to the original nine empirical and three simulated (n = 1,000) long‐term waterborne microbial count data sets. Both developments result from a theory of multiplicative growth of outcome size from correlated, entropy‐forced cause magnitudes. The predicted DW and DGD are first borne out in simulations of continuous and discrete correlated growth processes, respectively. Then the DW and DGD are each demonstrated to fit 10 of the original 12 data sets, passing the chi‐square goodness‐of‐fit test (α= 0.05, overall p = 0.1184). The PLN was not demonstrated, fitting only 4 of 12 data sets (p = 1.6 × 10−8), explained by cause magnitude correlation. Results bear out predictions of monotonically decreasing distributions, and suggest use of the DW for inhomogeneous counts correlated in time or space. A formula for computing the DW mean is presented.

Suggested Citation

  • James D. Englehardt & Ruochen Li, 2011. "The Discrete Weibull Distribution: An Alternative for Correlated Counts with Confirmation for Microbial Counts in Water," Risk Analysis, John Wiley & Sons, vol. 31(3), pages 370-381, March.
  • Handle: RePEc:wly:riskan:v:31:y:2011:i:3:p:370-381
    DOI: 10.1111/j.1539-6924.2010.01520.x
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    References listed on IDEAS

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    1. James D. Englehardt, 2002. "Scale Invariance of Incident Size Distributions in Response to Sizes of Their Causes," Risk Analysis, John Wiley & Sons, vol. 22(2), pages 369-381, April.
    2. James Englehardt & Jeff Swartout & Chad Loewenstine, 2009. "A New Theoretical Discrete Growth Distribution with Verification for Microbial Counts in Water," Risk Analysis, John Wiley & Sons, vol. 29(6), pages 841-856, June.
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    Cited by:

    1. Riccardo Patriarca & Tianya Hu & Francesco Costantino & Giulio Di Gravio & Massimo Tronci, 2019. "A System-Approach for Recoverable Spare Parts Management Using the Discrete Weibull Distribution," Sustainability, MDPI, vol. 11(19), pages 1-15, September.
    2. Katherine M. Anderson & Kevin Dayaratna & Drew Gonshorowski & Steven J. Miller, 2022. "A New Benford Test for Clustered Data with Applications to American Elections," Stats, MDPI, vol. 5(3), pages 1-15, August.
    3. Vegard Nilsen & John Wyller, 2016. "QMRA for Drinking Water: 2. The Effect of Pathogen Clustering in Single‐Hit Dose‐Response Models," Risk Analysis, John Wiley & Sons, vol. 36(1), pages 163-181, January.
    4. Barbiero, A., 2019. "A bivariate count model with discrete Weibull margins," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 156(C), pages 91-109.

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