Order Statistics and Benford's Law
Joint Authors
Nigrini, Mark J.
Miller, Steven J.
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2008, Issue 2008 (31 Dec. 2008), pp.1-19, 19 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2008-12-01
Country of Publication
Egypt
No. of Pages
19
Main Subjects
Abstract EN
Fix a base B>1 and let ζ have the standard exponential distribution; the distribution of digits of ζ base B is known to be very close to Benford's law.
If there exists a C such that the distribution of digits of C times the elements of some set is the same as that of ζ, we say that set exhibits shifted exponential behavior base B.
Let X1,…,XN be i.i.d.r.v.
If the Xi's are Unif, then as N→∞ the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior.
If instead Xi's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of any Nδ consecutive differences and all N−1 normalized differences of the order statistics exhibit shifted exponential behavior.
We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.
American Psychological Association (APA)
Miller, Steven J.& Nigrini, Mark J.. 2008. Order Statistics and Benford's Law. International Journal of Mathematics and Mathematical Sciences،Vol. 2008, no. 2008, pp.1-19.
https://search.emarefa.net/detail/BIM-467730
Modern Language Association (MLA)
Miller, Steven J.& Nigrini, Mark J.. Order Statistics and Benford's Law. International Journal of Mathematics and Mathematical Sciences No. 2008 (2008), pp.1-19.
https://search.emarefa.net/detail/BIM-467730
American Medical Association (AMA)
Miller, Steven J.& Nigrini, Mark J.. Order Statistics and Benford's Law. International Journal of Mathematics and Mathematical Sciences. 2008. Vol. 2008, no. 2008, pp.1-19.
https://search.emarefa.net/detail/BIM-467730
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-467730