Reviewed results with my adviser yesterday and the verdict is "go". Biggest task is framing the result in the context of current research. To that end, here's my bibliography so far:
Why heavy tailed distributions show up and why they are a problem:
Mandelbrot, Benoit. “The Pareto-lévy Law and the Distribution of Income”. International Economic Review 1.2 (1960): 79–106. Web...
Nolan, J.P. Stable Distributions — Models for Heavy Tailed Data. Birkhauser, Boston, 2015. Note: In progress, Chapter 1 online at academic.2.american.edu/~jpnolan. This is a book that is still a work in progress, but just ch1 is good background info. The author has a good page on Stable Distributions.
No Stable Distributions in Finance, Please! I'm not sure if this one's been published or not. It seems to raise some valid points about the overuse of Stable distributions. Worth a read just because the English is comically bad (certainly better than my Czech, but it's still funny)
L. B. Klebanov, G. M. Maniya, and I. A. Melamed, A Problem of Zolotarev and Analogs of Infinitely Divisible and Stable Distributions in a Scheme for Summing a Random Number of Random Variables Theory Probab. Appl., 29(4), 791–794. I couldn't find the full text online at any site I have access to so I'll have to track down a hard copy in a library. If I can't get this one in St. Louis, I know I can read it at Cornell when I visit my folks in June because we used to have this journal in the OR department.
Stratified sampling:
Anderson, Paul H.. “Distributions in Stratified Sampling”. The Annals of Mathematical Statistics 13.1 (1942): 42–52. Classic background on stratified sampling.
Shahrokh Esfahani, Mohammad; Dougherty, Edward R. (2014). "Effect of separate sampling on classification accuracy". Bioinformatics 30 (2): 242–250. Discusses bias issues arising from stratified sampling when the ratio of data between strata is not known.
Two texts I should probably read; hopefully I can find them in a library cuz they're REALLY EXPENSIVE. Fortunately, my family is in Ithaca and Kate's is in Champaign, so I'm regularly making trips that land me near two of the world's best mathematics libraries.
Heavy-Tailed Distributions in VaR Calculations. Chapter in Springer Handbooks of Computational Statistics.
Svetlozar T. Rachev & Stefan Mittnik, Stable Paretian Models in Finance
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