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CATEGORIES:Probability Theory and Statistics in High and Infi
nite Dimensions
SUMMARY:The Hurst Phenomenon and the Rescaled Range Statis
tic - David M. Mason\, University of Delaware
DTSTART;TZID=Europe/London:20140625T093000
DTEND;TZID=Europe/London:20140625T100000
UID:TALK53111AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/53111
DESCRIPTION:In 1950 H. E. Hurst published the results of his i
nvestigations of water out\now from the\ngreat lak
es of the Nile basin. Hurst wanted to determine th
e reservoir capacity that would\nbe needed to deve
lop the irrigation along the Nile to its fullest e
xtent. His work motivated\nthe notion of long rang
e dependence through the application of a statisti
c that he developed\nfor his study. This is the re
scaled range statistic-the R=S statistic.\nGiven d
ata 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) : Dene the adjusted range\nRn\n= M\nn
m\nn: The rescaled range statistic (the R=S stat
istic) is Rn\n= M\nn m\nn: where Dn\nis the sa
mple standard deviation Dn =\nqPn\ni=1 (Xi n)2
=n\, for n 1:\nHurst argued via a small simulati
on study that if Xi\, i = 1\; : : : \; n\, are i.i
.d. normal then\n(R=S)n should grow in the order o
f\np\nn. (Hurst was later proved correct by Feller
.) However\,\nHurst found that for the Nile River
data\, (R=S)n increased not in the order of\np\nn\
; but in\nthe order nH\, where H ranged between :6
8 and :80 with a mean of :75. For annual tree\nrin
g data H ranged between :79 and :86 with a mean of
:80\, for sunspots and wheat prices\nan average H
of :69 was obtained\, and data on the thickness o
f annual layers of lake mud\ndeposits gave an aver
age of H = :69. All of the above data had normal-l
ike histograms\, yet\nall gave estimates of H cons
istently greater than 1=2\, which an i.i.d. normal
model would\ngive. This is now called the Hurst p
henomenon.\nWe shall discuss some unexpected unive
rsal asymptotic properties of the R=S statistic\,
which\nshow conclusively that the Hurst phenomenon
can never appear for i.i.d. data.
LOCATION:Centre for Mathematical Sciences\, Meeting Room 2
CONTACT:
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