Chapter 10: Complementing Benford’s Law for small N: a local bootstrap (Roukema)

• Additional Reading:
  • B. Efron, Bootstrap Methods: Another Look at the Jackknife, Annals of Statistics, 7, 1, 1979, open access: http://projecteuclid.org/euclid.aos/1176344552.
  • B. F. Roukema, A first-digit anomaly in the 2009 Iranian presidential election, Journal of Applied Statistics, 41, 164, 2014; http://dx.doi.org/10.1080/02664763.2013.838664 or
    open access preprint http://arXiv.org/abs/0906.2789v4.
• Homework Problems:
  • In addition to the ones in the book:
    •  Direct first-round multi-candidate presidential elections in countries with large populations, at least 100 to 300 electoral divisions of varying population sizes, and online open access to the official results, have occurred since the Iranian 2009 election in (at least) Indonesia (2009), France (2012), Iran (2013), Poland (2010), and Indonesia (2014). Find the online sources. Download the data, if necessary using GNU/Linux command line tools like ``wget'' while including automatic delays in order not to load the servers too heavily. Check whether these results show any local bootstrap first-digit frequency anomalies, using the earlier calibration of the method [Rou].
    • Find data for an ecological habitat in which species (e.g. plants, animals, fungi, or bacteria) compete with each other, and for which a field team has published the full data of its claims of
      having counted the number \(v_{ij}\) of members of the \(j\)-th species in the \(i\)-th geographical region, for all \(i\) regions and \(j\) species. The fractional population \(w_{ij} := v_{ij}/x_i\) of each species competing for a finite amount of resources varies among the regions, corresponding to the voting rate in the Iranian election example. Check that the logarithmic spreads \(\sigma(w_{ij})\) are high enough for this approach to be useful (cf Fig.~10.1). Generate local bootstrap simulations (Defn~10.3.1, Eqs~(10.2), (10.3)) and confidence intervals for the first-digit frequencies of the species' numbers. Plot the first-digit frequencies of the \(v_{ij}\), the local bootstrap confidence levels, and the Benford's Law limit, as in Fig.~10.3. Plot the folded logarithmic distributions as in Fig.~10.4. Is an anomaly found? If yes, does further evidence support the case for an anomaly, or does the subset of the data traced by the anomaly statistically resemble the complementary subset?
• Videos:
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