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SUMMARY:The Hurst Phenomenon and the Rescaled Range Statistic - David M. M
 ason\, University of Delaware
DTSTART:20140625T083000Z
DTEND:20140625T090000Z
UID:TALK53111@talks.cam.ac.uk
CONTACT:37296
DESCRIPTION:In 1950 H. E. Hurst published the results of his investigation
 s of water out\now from the\ngreat lakes of the Nile basin. Hurst wanted t
 o determine the reservoir capacity that would\nbe needed to develop the ir
 rigation along the Nile to its fullest extent. His work motivated\nthe not
 ion of long range dependence through the application of a statistic that h
 e developed\nfor his study. This is the rescaled range statistic-the R=S s
 tatistic.\nGiven data Xi\, i = 1\; : : : \; n\, set n = 1\nn\nPn\ni=1 Xi\
 ; and let S\ni =\nPi\nj=1 (Xj 􀀀 n)\, for i =\n1\; : : : \; j\; M\nn 
 = max (0\; S\n1 \; : : : \; Sn\n) and m\nn = min (0\; S\n1 \; : : : \;
  Sn\n) : Dene the adjusted range\nRn\n= M\nn 􀀀m\nn: The rescaled r
 ange statistic (the R=S statistic) is Rn\n= M\nn 􀀀m\nn: where Dn\nis
  the sample standard deviation Dn =\nqPn\ni=1 (Xi 􀀀 n)2 =n\, for n  1
 :\nHurst argued via a small simulation study that if Xi\, i = 1\; : : : \;
  n\, are i.i.d. normal then\n(R=S)n should grow in the order of\np\nn. (Hu
 rst was later proved correct by Feller.) However\,\nHurst found that for t
 he Nile River data\, (R=S)n increased not in the order of\np\nn\; but in\n
 the order nH\, where H ranged between :68 and :80 with a mean of :75. For 
 annual tree\nring data H ranged between :79 and :86 with a mean of :80\, f
 or sunspots and wheat prices\nan average H of :69 was obtained\, and data 
 on the thickness of annual layers of lake mud\ndeposits gave an average of
  H = :69. All of the above data had normal-like histograms\, yet\nall gave
  estimates of H consistently greater than 1=2\, which an i.i.d. normal mod
 el would\ngive. This is now called the Hurst phenomenon.\nWe shall discuss
  some unexpected universal asymptotic properties of the R=S statistic\, wh
 ich\nshow conclusively that the Hurst phenomenon can never appear for i.i.
 d. data.
LOCATION:Centre for Mathematical Sciences\, Meeting Room 2
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