Generalizations of power-law distributions applicable to sampled fault-trace lengths: model choice, parameter estimation and caveats

Abstract

Summary It has often been observed that fault‐trace lengths tend to follow a power‐law or Pareto distribution, at least for sufficiently large lengths. A very common method of fitting this type of model to data consists of plotting on log–log axes the number of faults with trace length greater than x against x, and reading off the slope of the resulting approximate straight line. We demonstrate that maximum likelihood is a more efficient and less biased method of estimating the power‐law exponent. A further complication is that this log–log plot is often curved, suggesting that the power‐law distribution is not a complete description of the data. In this paper we review the literature on probability distributions with Pareto behaviour for long trace lengths, but not necessarily for short trace lengths. The Feller–Pareto distribution is an attractive family within this class, with many well‐known statistical distributions as special cases. We use maximum likelihood to fit the Feller–Pareto distribution to a sample of 1034 fault‐trace lengths from the South Yorkshire coalfields. We conclude that the Burr III model superficially provides a satisfactory fit to these data. We also discuss an interpretation of the Feller–Pareto model in terms of a particular type of observational bias on data generated from the power‐law distribution. However, there are a number of complications to be considered. In particular, geometrical sampling biases, stereological effects and spatial structure in the data mean that a rigorous analysis is not straightforward. We suggest ways in which future data collection and analysis may address some of these problems. If our sampling protocols and estimation procedures are adopted, geoscientists should be able to estimate the power‐law exponent more accurately and more objectively than with current ad hoc procedures, and with more direct relevance to strain calculations and other geophysical applications. Furthermore, our recommended method of estimation, maximum likelihood, provides point estimates and associated standard errors of the unknown parameters, and is efficient, consistent and relatively straightforward to apply.

On Estimating the Exponent of Power-Law Frequency Distributions

“…Because maximum likelihood estimation for the truncated Pareto requires numerical methods, it has been suggested that in some cases with both a minimum and maximum value that the error introduced by assuming that there is no maximum is small enough that it is reasonable to estimate the exponent using the maximum likelihood estimate for the Pareto distribution. Clark et al (1999) suggest this approximation in cases where the maximum value is at least two orders of …”

“…There is an error in the MLE solution given by Evans et al (2000) that has been corrected. Note that MLEs are only guaranteed to be minimum variance unbiased estimators in the limit of large n. If n is small, corrections to the MLE are available (Johnson et al 1994, Clark et al 1999, Clauset et al 2007). All solutions assume that a and b are known.…”

“…Alternatively, the likelihood can be maximized directly using numerical methods (Clauset et al 2007, Zillio andCondit 2007). While MLE does not provide an opportunity for visual inspection of the distribution to determine if the assumption of the power-law functional form is reasonable, the validity of this assumption can be assessed using simple goodness-of-fit tests such as the Chi-square on binned data (Clark et al 1999, Clauset et al 2007, Edwards et al 2007), or by visually assessing the linearity of binned data, or the CDF (Benhamou 2007), under the appropriate transformation.…”

“…2; Clark et al 1999, Newman 2005. This probably results because the logarithmic transformation used in fitting the CDF weights a small number of points more heavily, and because the points in the CDF are not independent thus violating regression assumptions (see Clauset et al [2007] for other issues with regression-based approaches).…”

“…In some cases deviations may suggest that the power law is in fact not the appropriate model for the data. This can be evaluated using goodness-of-fit tests on binned data (Clark et al 1999, Clauset et al 2007, Edwards et al 2007) or by using model selection techniques to compare the power-law to alternative distributions (Muller-Landau et al 2006, Clauset et al 2007, Edwards et al 2007.…”