{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,13]],"date-time":"2026-06-13T07:26:29Z","timestamp":1781335589814,"version":"3.54.1"},"reference-count":32,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2008,12,1]],"date-time":"2008-12-01T00:00:00Z","timestamp":1228089600000},"content-version":"vor","delay-in-days":335,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-0600848"],"award-info":[{"award-number":["DMS-0600848"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["onlinelibrary.wiley.com"],"crossmark-restriction":true},"short-container-title":["International Journal of Mathematics and Mathematical Sciences"],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>Fix a base<jats:italic>B<\/jats:italic>&gt; 1 and let<jats:italic>\u03b6<\/jats:italic>have the standard exponential distribution; the distribution of digits of<jats:italic>\u03b6<\/jats:italic>base<jats:italic>B<\/jats:italic>is known to be very close to Benford\u2032s law. If there exists a<jats:italic>C<\/jats:italic>such that the distribution of digits of<jats:italic>C<\/jats:italic>times the elements of some set is the same as that of<jats:italic>\u03b6<\/jats:italic>, we say that set exhibits shifted exponential behavior base<jats:italic>B.<\/jats:italic>Let<jats:italic>X<\/jats:italic><jats:sub>1<\/jats:sub>, \u2026,<jats:italic>X<\/jats:italic><jats:sub><jats:italic>N<\/jats:italic><\/jats:sub>be i.i.d.r.v. If the<jats:italic>X<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2032s are Unif, then as<jats:italic>N<\/jats:italic>\u2192<jats:italic>\u221e<\/jats:italic>the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If instead<jats:italic>X<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>\u2032s come from a compactly supported distribution with uniformly bounded first and second derivatives and a second\u2010order Taylor series expansion at each point, then the distribution of digits of any<jats:italic>N<\/jats:italic><jats:sup><jats:italic>\u03b4<\/jats:italic><\/jats:sup>consecutive differences<jats:italic>and<\/jats:italic>all<jats:italic>N<\/jats:italic>\u2212 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\u2032s law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.<\/jats:p>","DOI":"10.1155\/2008\/382948","type":"journal-article","created":{"date-parts":[[2008,12,2]],"date-time":"2008-12-02T16:21:57Z","timestamp":1228234917000},"update-policy":"https:\/\/doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Order Statistics and Benford\u2032s Law"],"prefix":"10.1155","volume":"2008","author":[{"given":"Steven J.","family":"Miller","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark J.","family":"Nigrini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"311","published-online":{"date-parts":[[2008,12]]},"reference":[{"key":"e_1_2_6_1_2","first-page":"551","article-title":"The law of anomalous numbers","volume":"78","author":"Benford F.","year":"1938","journal-title":"Proceedings of the American Philosophical Society"},{"key":"e_1_2_6_2_2","doi-asserted-by":"publisher","DOI":"10.1511\/1998.31.815"},{"key":"e_1_2_6_3_2","doi-asserted-by":"publisher","DOI":"10.2307\/2319349"},{"key":"e_1_2_6_4_2","unstructured":"HurlimannW. 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