A Semiparametric Bootstrap Technique for Simulating Extreme Order Statistics Author(s): Daniel Zelterman Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 88, No. 422 (Jun., 1993), pp. 477485 Published by: American Statistical Association Stable URL: http://www.jstor.org/stable/2290327 . Accessed: 25/07/2012 08:02 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp
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A Semiparametric Bootstrap Technique forSimulating ExtremeOrder Statistics DANIELZELTERMAN* We proposea techniqueforsimulatingthejoint distribution of thej largestorderstatistics of a verylargesample.We assumethat the parentpopulationis in thedomain of attractionof theType I (Gumbel) extremevalue distribution. The bootstrapvariatesare generatedby resamplingthe normalizedspacingsof the k largestobservedvalues in the originaldata wherek is largerthanj. We comparethebootstrap distribution to thefitted extremaldistribution ofWeissman.Bothdistributions havethesamemeans,conditional on the k largestobservedvalues in the data set. If k is largeand the normalizedspacingsbehave as independentand identically distributed exponentialrandomvariablesthenthebootstrapvariatesbehaveas thoughsampledfromtheextremaldistribution. We proposeseveralproceduresforestimating k and givea numericalexample. KEY WORDS:
Exponentialdistribution; Extremevalue distribution; Gini statistic;Spacings.
1. INTRODUCTION
distributionfunctionF( *). This articleis concernedwith developinga bootstrapapproximationto the joint distributionof the k largestorderstatisticsfromF whenF( *) is unknownand n is muchlargerthank. Bickeland Freedman (1981, sec. 6) remarkedthatsimplyresamplingtheoriginal data is a bad idea. Clearly,theusual bootstraptechniqueof resamplingthe observeddata will not work,because it is impossibleto obtainpseudosampleswithvaluesgreaterthan X1,. Even forlargesamplesizes,thenaive bootstrapforthe largestorderstatisticwill have a large point mass at XIn. Instead,we proposeresamplingthesamplespacings,or differencesbetweenadjacentorderstatistics. When properlynormalized,thespacingsofa smallsetof thelargestorderstatistics frommanycommonlyencountered distributionsbehave as iid exponentialrandom variables whenthesamplesize is large.The proposedmethodresamples thesenormalizedspacings,"unnormalizes"theirbootstrappedvalues,and thenadds theseback togetherto simulatethebehavioroflargeorderstatistics. An earlierversion ofthistechniqueappearedin a conference proceedings(Zeltermanand Lindgren1992). The details of the bootstrap techniqueare given in the followingsection.Beforethat, however,we mustdescribetheasymptotic behaviorofspacingsin greaterdetail. We will assume thatF(*) is in the domain of attraction ofthe Type I (Gumbel) extremevalue distribution denoted by G(x) = exp(-e-x) for-oo < x < oo. Specifically, this meansthatthereexistsequencesofrealnumbersan> 0, bn, n = 1, 2,... such thatforall real values ofx,
The studyofextremeorderstatistics has longbeen a concernofstatisticians and is becomingincreasingly important withits applicationto environmental and public healthissues. A highrate of cancer or a major floodare headlines thatattractpublicattention, nottheaveragerainfall or typical tumorrisk.The Dutch government, forexample,has legislatedthatleveesand sea dikesmustwithstanda 1 in 10,000 chance of failure.This has provokeda seriesof statistical studiesintowhatheightis adequate protection (Dekkersand de Haan 1989). Unusuallyhighratesofchildhoodleukemia or birthdefectsare cause foralarm (Lagakos, Wessen,and Zelen 1986). Otherexamplesare theestimationofthemaximumconcentration ofairbornepollutantsin a metropolitan area (Smith 1989) and the maximumexposureto radioactivity releasedbya nuclearpowerplant(Davidsonand Smith 1990). The bootstrapis a flexibletechniquethatcan be applied to a wide varietyofproblems.A good introduction is Efron and Gong (1983) orthemonographbyEfron(1982). Recent reviewsincludeHinkley(1988) and Diciccio and Romano (1988). The bootstrapallowsus to obtainapproximatesamples fromthe distributionof a random variable without a parametricformforthe distribution. specifying The usefulnessof this abilitycannot be overestimated: the distribution,significance level,bias,variance,and so forthcan all be approximatedin a nonparametricframework.Recent workon quantileestimationusingbootstraptechniquesinclude Davidson (1988), Johns(1988), and Do and Hall (1991). None of these references gave special attentionto lim Fn((x - b)/an) = G(x). (1.1) n-cx3 extremequantiles. Major drawbackswiththebootstiapare therequirements In words,whenXI n is properlynormalized,itbehavesas an thatthe parentdata sample representindependent,identi- observationfromthe distributionG forverylargesample cally distributed(iid) random variablesand that the sta- sizes,n. This assumptionis not overlyrestrictive and is sattisticunder studybe symmetricin its arguments.Let XI, isfiedby manycommonlyencountereddistributions, such > X2, . . . > X,, denotethe descendingorderstatisticsof a as thenormal,lognormal,gamma,Pareto,Gompertz,Weirandomsampleofsize n froma populationwithcumulative bull, and the logisticdistributions as well as the Gumbel extremevalue distributionitself-but not the uniformor * Daniel Zeltermanis AssociateProfessor, forexample. DivisionofBiostatistics, School Cauchy distributions, of PublicHealth,University of Minnesota,Minneapolis,MN, 55455. This researchwas supportedby National Institutesof Health Grants P01AG0876 1,UOI -AG10328,P0 I -DCOO133, and NCI-AI05073and bya grant fromtheMinnesotaSupercomputer Institute. The authorthanksT. A. Louis, J. Wang, R. D. Cook, and J. W. Vaupel fortheirhelpfulcommentsand sumestions-
477
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June 1993. Vol. 88. No. A22. Theorv and Methods
Journal of the American Statistical Association, June 1993
478
on thevaluesofthefitted parameters a and David(1981),Serfling (1980),andReiss(1989)aregeneral Conditional twomoments ofXjk are Muchoftherecentliteratureb, thefirst references on orderstatistics. is concerned withestimating the E(Xjla, b) = dE(Mj) + b = a(Sk-Sj-l on extreme orderstatistics ) +X(k+l)n (1.5) formofthelimitofFn((x - bn)/an).The genfunctional introduced eralizedParetodistribution byPickands(1975) and ofa distribution. var(XjIa, b) = a2var(Mj) tailbehavior is often usedtomodelextreme tothisdistribution. Other Smith(1989)isa goodintroduction oo j-1 tailofa distribution aregiveninHill modelsoftheextreme = a2 z i-2 = a2r2/6i-2 , (1.6) (1975) and Zelterman (1992). The pointhereis thatthere i i=j in (1.1), butthis distributions are otherpossiblelimiting = 0 and So = 0. Becausetheexactform whereE? is notoverly restrictive. assumption thebootstrap WewilluseLemma1tomotivate byresambutonlytheshapeofitsextreme ofF is notspecified, tail, In Section iidnormalized plingtheapproximately spacings. to ourtechnique as semiparametric. werefer indetailandcompare thebootstrap technique than 2 wedescribe Forsomefixedvalueofk (=2, 3, .. .) muchsmaller totheconditional distribution ofthebootstrap the moments onthejointasymptotic behavior ofthe n,letus concentrate of moments Xjk givenin (1.5) and (1.6). In Section3 we in a verylargesampleofsizen. k + 1 largest orderstatistics behave kis andthenormalized spacings and arefoundin showthatwhen large results arewellestablished The following the then distributions random iid as exponential variables, threelemmasfollow Weissman(1978). The following coincide.In Section4 we variates ofXjk and thebootstrap from(1.1). theappropriate valueofk. forchoosing discussprocedures with numerical studies 5 some In we conclude Section using Lemma1 (Weissman1978,thm.3). For fixedk, as n real data. oo, the normalizedspacingsian(Xin - X(?i)n) (i = 1, ...
,
k) areasymptotically jointlydistributed as independent
standard randomvariables. exponential
Lemma 2 (Weissman 1978, thm.2). As n -* oo, the limitingjoint behaviorofMjn= (Xjn-bn)/anforj = 1,. . .
2. THEBOOTSTRAPAND ITS CONDITIONALMOMENTS
? Xn, ofa random Giventheorderstatistics Xln 2 ... kisthek-dimensional extremal with distribution jointdensity samplefrom a population, wewillconstruct bootstrap pseufunctiong (ml, ... Mk) = exp { -emk - EL= mj} forml dosamplesofthek largest ofthese.(The secondindex,n, ? M2 .** > Mk. The asymptotic func- willbe omittedbut understood marginal density throughout.) Throughout tion ofMin (j = 1, - -* k) is thissectionk is fixedandmuchsmallerthann. Definethenormalized spacingsd = { di} ofthek + 1 (1.2) largest gj(m) = exp {-e-m- jm }/(j- 1)!, orderstatistics by
defined on -oo < m < oo.
di = i(Xi-X,X+)
i= 1,...k.
(2.1)
Lemma 3 (Weissman1978, thm.5). The minimum Whennis muchlarger thank,Lemma1 saysthatthediare ofanandbn,basedonthejoint approximately variance unbiasedestimators iid exponential randomvariables. The defiof { Xl n,X* X(k+ i)n } givenbyg* (*) nitionof"normalized asymptotic distribution in is different spacings" (2.1) slightly in Lemma2, are in Lemma1,becausewe do notknow fromthedefinition = ittoperform thesimulation. an anddo notneedtoestimate a ak Xk - X(k+1)n (1.3) a bootstrap of Let { dl, . d* } represent pseudosample = = = k from ) size drawn with d. b The replacement and bootstrap pseubk + X(k+l)n, whereXk k ak(Sk+l X i Xi, Sk = Xk i(1, andy = .5772 .... orderstadosampleX * > X 2 > ... > X * ofthek largest is defined by An approximation to theasymptotic distribution ofXln tistics k can easilybe constructed usingLemmas2 and 3. Foreach = 1,...,k). i-'d*i XJ i =Xk+l ~+ (2.2) j = 1,. . . k, define therandomvariables ~~~~ Xj = Xjk=-akMj + bk,
(1.4)
i=j
Thatis,X* = Xk+1+ d*/k and XJ*= XJ*1+ j-1 dj for whereMj is a randomvariablewithdensity function gj( ) j = k - 1, ..., 2, 1. Asterisksalways indicatebootstrap givenat (1.2). Whenn is large,wewillconsider usingXjk as variates,and the tilde (-) refersto the parametric a parametric modelforthebehaviorofXjn. Boos (1984) model(1.4). AlusedXjas a parametric behindthistechniqueis as follows. modeltoestimate extreme quantiles The motivation ofF. Thisarticle onapproximating X1> X2> ... arejointly theoriginal orderstatistics concentrates themarginal though ton) set distributions of each XJn(j = 1, . . . k). We could just as dependent, thata small(relative Lemma1 suggests whenproperly areapproximately easilysamplethe{ Mj } jointlyfromg and usethe{ Xjn} to ofk spacings, normalized, result thatthesespacings approximate thejointbehavior of{ Xjn} . Notethatg1is the iid.We do notneedtheadditional densityfunction of theGumbeldistribution thatappears arealsoapproximately exponentially distributed togenerate in (1.1). thebootstrap variates.NotethatXk?1 staysfixedin (2.2)
479
Zelterman: Bootstrapping Extreme Order Statistics
thesimulation.When k is moderatelylarge,the throughout varianceofXI is muchgreaterthanthevarianceof Xk+1, so thereis fairlywide latitudein thechoiceofk. Some suggestionsforchoosingk are givenin Section4. The remainderofthissectionis concernedwithmoments of XJ conditional on the normalized spacings d and functionsof X7 Xk?,. We give the moment-generating X X . . ., We then look at of jointly. I, marginallyand k theunconditionalvarianceof XJ*,assumingthatthe { di } In the next are independentand exponentiallydistributed. of XJ function the moment-generating sectionwe expand of k. values forlarge To findmomentsof X for] = 1, ... k, note thatXJ has the equivalentdefinition k
k
XJ = Xk+1+ k i-l
E
r=I
i=j
(2.3)
nr(i)dr.
functionof XJ is The conditionalmoment-generating E{exp(tX
k
=
E(X
Xk+I
+
k E
i=J
X,...,X
E{exp J=J
=
dr= (XI -X2)+
Xk+?) =
= k(Xk-
exp(tdr/i)
(2.7)
d, Xk+l}
exp(t1dr)}
E exp((t1 + t2)dr/2)}
dr}
E
wheret. = Ej
*+ k(Xk-Xk+1)
{k-1 E exP(t.dr/k)},
> k i-l
t..
In particular,for1 < j' < j < k, cov(X*,
XJ [d, Xk+I) = var(XjI d, Xk+?) and corr(X3,, { Zl
(2.4)
kal,
whered wasgivenin (1.3). ThenforSk =
tjX)
X {k-
*
2(X2 X3)+
k
r=l
exp(t.Xk+?) X {k-1
k
r=I
k-1 r
k iS
By (2.1), notethat E
17
function moment-generating using(2.3) and themultinomial (Bishop,Fienberg,and Holland 1975,p. 441). In thefollowas k - oo to ing sectionwe expand (2.7) asymptotically thelimitingbehaviorof thebootstrapvariates demonstrate XJ . An Edgeworthexpansioncould also be derivedfrom but we will not do (2.7) and this asymptoticdistribution, this. functionof The joint conditional moment-generating
r
i-lk-j '
exp(tXk+l)
I=1
The n(i) = (n(i), . . nk) in (2.3) are independentmulti= 1 nomial vectorsfori = 1, ..., k, each with Erk= and all probabilityvectorsequal to (k-1, ... k-1). Using (2.3), we can derivemomentsof X* conditionalon (d, Xk+l). In particular, Id, Xk+1) =
)Id, Xk+1}
i-2/Z=
XJ)
, i-2 }1/2
We end thissectionby showingthatunconditionalon d,
andevery thedifference betweenthevariancesofXJ and XJ, givenin
(1.6) and (2.6), becomes smallerforlargervalues of k. (In the followingsectionwe demonstratethatthe distributions (2.5) E(Xj Id, Xk+1) = d(Sk- Sjft) + Xk+,, of XJ*and XJcoincide when k is large.)Assume thatthe a random limitin Lemma 1 holds;thatis, thatd represents whichis the same as the conditionalmean of Xj givenin withknownmean samplefroman exponentialdistribution (1.5). That is, the bootstrapvariateXJ has the same con> 0. Then by (2.4), ditionalmean as the parametricmodel Xj giventhe k + 1 a, . largestobservedorderstatistics XI >_ Xk+I (2.8) Ea = Ek-' E dr= a, Continuingfrom(2.3), we have (2.9) var a = a2/k, var(X* I|d,Xk+I) k k and Ek1-1 (dr - d)2 = a2(k - 1)/k. i-2var E n(I)drd) Using (2.5) and (2.6), the unconditional(on d) variance r=l i=J of XJ is k
k
[k-2(k = i-2~ I=j r=k1r'r
- l)d]2-
k k-2
dr
.
= var{E(X,
Using (2.4) gives var(X Id,Xk+,)
k
Xk =
(k
i=J
var(X* IXk+1)
k2)l-l
E (dr-
r=l
d)2.
(2.6)
Attheend ofthissectionwe compare(2.6) to theconditional varianceofXjk givenin (1.6).
Id, Xk+l) Xk+l} + E{var(X, ld, Xk+l)}
J~
= var{a-(Sk-Sjl)}+
~
J
i=j
i2)E{k-1
(dr -)2}
Journal of the American Statistical Association, June 1993
480
The analogousresultfrom(1.5) and (1.6) is
means a > 0. Using (1.5), (2.5), and (2.8), both Xj and X7 have the same conditionalmeans equal to a log k + Xk+1 + 0(1) givenXk+1.The marginalvariancesof XJ and Xj, givenin (2.10) and (2.11), areboth 0(l) as k oo forfixedvalues ofJ. We willconsiderthelimitingformsof themoment-generating functionsofXj and XJ)whenproperlynormalizedby thesemoments. For fixedj = 1, 2 ..., much smallerthan k and all t sufficiently close to 0, define
var(XjIXk+I) = var{E(XIda, b)IXk+l} = var{a(Sk
- Sj,-)}
= a2k-1{(Sk
+
E{var(X,Ia,
+ (X2/6
-
Ei-2
b)} E2
- Sj_I)2
+ (k + 1) (2 7
/6-
Mj(t) = E exp[t(Xj - a log k - Xk+?)]
(.1)
i-2)
= EE{exp[t(Xj-
alogk-Xk+l)]Id,Xk+l}
The unconditionalbootstrap variance (2.10) is slightly = E exp[dt(Sk- y) - at logk]r(j - at)/r(j), smallerthantheunconditionalvariance(2.1 1) predictedby themodelXj. Foranyfixedvalueofj, however, thedifference makinguse of (1.2) and (1.4). Referto (2.4) and note that between(2.10) and (2.1 1) usuallybecomes small as k be- a has a gamma distribution. Use (2.8), (2.9), and Sk - y comes large.In particular, = log k + 0(k-1) (Jolley1961, p. 14) to showthat var(XjIXk+l) - var(X* IXk+1)
at)/r(j) }[ i+
mj(t) = {r'(j-
0 (k2log
k)]
(3. 1)
That is, when k is large,Xj given Xk+l has approximately thesame distribution as a(Mj + log k) + Xk+1, whereMj is i=j a random variable with the densityfunctiongj(.) given O(a2k-1). in (1.2). The moment-generating functionof Xj*, properlynorIn most commonly encounteredsituations,this last expressionwilltendto 0 at leastas fastas k-'. For example, malized,is foundfrom(2.7). Define in the normaldistributionan = (2 log n) -1/2, and in the Mj*(t) = E exp[t(X -a log k - Xk+j)I logisticand exponentialdistributions an = 1 (Leadbetter, Lindgren,and Rootzen 1983,p. 20). = EE{exp[t(X -alog k-Xk+l)] Jd, Xk+l} - 2ak1
, i2 + a2(l -
1)( 6
3. THEUNCONDITIONAL ASYMPTOTIC BOOTSTRAPDISTRIBUTION
ii
)
=
k
k
i=j
r=1
k-atE 171k-1 z exp(tdr/i)}.
Here we show thatwhen k is largeand the normalized Then write spacingsd behave as iid exponentialrandomvariables,the k k ofXjand X J coincide.The proof unconditional distributions MJ (t) = k-at(1 (1 - at/i)-l EIl {I + bk(t/l)}, of thisassertionconsistsof comparingthe asymptoticmoi=j i=J functionsofXj and Xj7. Withsome addiment-generating where we mightalso comparethejoint moment-gentionaleffort, eratingfunctionofXj,... , XJwiththatof X 1,... X J for - (I - a .) (1 - aO){k1 I kexp(dr) k(O)= (3.2) some fixedvalue of J. This exercisewouldhelpdescribethe joint behaviorof the simulatedorderstatisticsbut would In the Appendixwe show thatE YJk=1 providelittleadditionalinsight. {I + bk(t/i)} = I In thissectionconsidera fixedvalue ofj = 1,2, . . . while + 0(k-/2log k), so that k -* oo. In practicethissettingoccurswhenthesamplesize k n is trulyhuge,so thata largenumber,k, of observations log MJ (t) = -at log klog(l - at/i) fallin the extremetail of the distribution, i=j but we are only concernedwiththebehaviorofa few(j) ofthelargestorder + 0(k-1/21ogk) statistics. Simulationsdescribedin thefollowing sectionshow = -at log k + log{r(k + l)/r(k + 1 - at)} thatifX1, . . . Xnare normallydistributed, forexample,then the use of k = nI 3 is not unreasonable. Dekkers + log{r(j - at)/F(j)} + 0(k-"/2log k). and de Haan (1989) consideredsequences k(n) satisfying in k log log n X6. locationof the six largestobservedvaluesXI > * Values of k much largerthan 50 would resultin bootstrap thelargestorderstatistics. variatesthatgreatlyunderestimate of XJ from(2.6) are deviations standard The conditional affected by overnot greatly are 4. These Figure in plotted k. estimating
6001l
0
400
(D
3~~~~~~~~~~~
100
0
Figure4. Standard Deviationsof the BootstrapVariatesX,(= ComputedFrom(2.6J.
1, ...6)
Conditionalon the k Largest OrderStatisticsin the Coal MiningData,
Journal of the American Statistical Association, June 1993
484
0.5
1.5
2.5
3.5
4.5
x 103 Figure5. FittedMarginalDensitiesof the ParametricModel X,(DottedLine) and Kernel-SmoothedDensities of the BootstrapVariatesX, (Solid Line) forj = 1, 2, 3 in the Coal MiningData. Thissimulationresampled the normalizedspacings of the k = 36 largestobservationsin the data. In ofXland X, coincide. Section 3 we show thatwhenk is large,the distributions
Figure5 givesestimatesofthedensitiesofthethreelargest The dottedlinesare thefittedmarginaldenorderstatistics. sityfunctionsof X,, X2, and X3, usingk = 36. The solid marginaldensityestimatesof linesare thekernel-smoothed
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