Abstract
We develop a variation of a Kolmogorov-Smirnov (KS) method for estimating a power law region, including its lower and upper bounds, of the probability density in a set of data which can be modelled as a continuous random sample. Our main innovation is to stabilize the estimation of the bounds of the power law region by introducing an adaptive penalization term involving the logarithmic length of the interval when minimizing the Kolmogorov-Smirnov distance between the random sample and the power law fit over various candidate intervals. We show through simulation studies that an adaptively penalized Kolmogorov-Smirnov (apKS) method improves the estimation of the power law interval on random samples from various theoretical probability distributions. Variability in the estimation of the bounds can be further reduced when the apKS method is applied to subsamples of the original random sample, and the subsample estimates are averaged to yield a final estimate.
2010 Mathematics Subject Classification:
Acknowledgments
PRK would like to thank the Mathematical Biosciences Institute at Ohio State University for support as a long-term visitor during its emphasis year on Stochastics in Biological Systems, during which this collaboration was initiated. We also thank Suzanne Ditlevsen for pointing out the relation of our proposed batching strategy to a technique used in spectral density estimation.
Disclosure statement
No potential conflict of interest was reported by the author(s).