【word】 混合指数分布顺序统计量的性质
混合指数分布顺序统计量的性质
第38卷第2期
2011年3月
浙江大学(理学版)
JournalofZhejiangUniversity(ScienceEdition)
ar.2011
DOI:10.3785/j.issn.1008—9497.2011.02.004
Onpropertiesoforderstatisticsfrom
l?-
themixedexDonentialdistribution
KUANGNeng—hui
(SchoolofMathematicsandComputingScience,HunanUniversityofScienceand
Technology,Xiangtan411201,HunanProvince,China)
Abstract:Let{X,1?s?n}beindependentandidenticallydistributed,andXlX2…,Xbethecorrespond—
ingorderstatistics.WhenXfollowsthemixedexponentialdistributionwithparametersp(O<户<1),l,A2(0<A1
?A2),theexplicitformulasfortheqorderoriginmomentsofXareobtained(qi
sapositiveintegerand1?5?
).Itisprovedthatthesampleintervalsoftheirorderstatisticsarenotindepende
ntandnotidenticallydistributed.
What’smore,theasymptoticdistributionsoftheirextremeorderstatisticsXl:
andX:arediscussed.
KeyWords:themixedexponentialdistribution;orderstatistic;originmomen
t;asymptoticdistribution
匡能晖(湖南科技大学
数学
数学高考答题卡模板高考数学答题卡模板三年级数学混合运算测试卷数学作业设计案例新人教版八年级上数学教学计划
与计算科学学院,湖南湘潭411201)
混合指数分布顺序统计量的性质.浙江大学(理学
版),2011,38(2):135—139
摘要:设{X,1??}独立同分布,XX.…,X…为其顺序统计量.当X服从
参数分别为P(O<p<1),,
(O<A?)的混合指数分布时,得到了X的q(q为正整数)阶原点矩
E(X:)(1?5?)的精确
表
关于同志近三年现实表现材料材料类招标技术评分表图表与交易pdf视力表打印pdf用图表说话 pdf
达式.证明了
其顺序统计量的样本间隔不独立,且不同分布.此外还研究了其极端
顺序统计量X:和…的渐近分布.
关键词:混合指数分布;顺序统计量;
中图分类号:02l1.4文献标志码
1Introduction
原点矩;渐近分布
:A文章编号:1008-9497(2011)02-135—05
Ar.v.Xissaidtohavethemixedexponential
distributionifitspdfis
厂cz==={?.二—2e,证z?.c
where0<<1and0<a1?2.Itiseasytogetits
edfis
…
f(1一e-)+(1一)(1一e-a2),ifz?0;F(
z)===c’l
0,ifz<0.
(2)
Themixedexponentialdistributionplaysan
importantroleinthelife—timedataanalysis.Many
researchersareinterestedintheparametersestima—
tionsofthemixedexponentialdistributionE1--3].
Themixedexponentialdistributionhasmanyprac—
ticalapplicationsinvariousfieldssuchasinsur—
ance,BayesianinferenceandServicesystems,
etcE一1
.
Momentsoforderstatisticsplayanimportant
roleinsuchareasasqualitycontroltestingandre—
liability.Ifthereliabilityofanitemishigh,the
durationofan”alliternsfail”life—testcanbetoo
expensiveinbothtimeandmoney.Thisfactpre—
ventsapractitionerfromknowingenoughabout
theproductinarelativelyshorttime.Therefore,a
practitionerneedstopredictthefailureoffuturei—
ternsbasedonthetimesofafewearlyfailures.
Thesepredictionsareoftenbasedonmomentsof
Receiveddate:November23.2009.
Foundationitem:TheProjectSupportedbyScientificResearchFundofHuna
nProvincialEducationDepartment(GrantNo.08C588)
Abouttheauthor:KUANGNeng—hui(1973
一),male,Master,thefieldofinterestincludesprobabilityandstatistics.
136浙江大学(理学版)第38卷
orderstatlStlCS.
Itiswellknownthatthesampleintervalsof
orderstatisticsfromtheuniformdistributionare
notindependentbutidenticallydistributed,and
thoseoforderstatisticsfromtheexponentialdis—
tributionareindependentbutnotidenticallydis—
tributedc.
Inthispaper,theexplicitformulasfortheori—
ginmomentsoforderstatisticsfromthemixedex—
ponentialdistributionareobtained.Itisproved
thatthesampleintervalsoftheirorderstatistics
arenotindependentandnotidenticallydistributed.
What’smore,theasymptoticdistributionsoftheir
extremeorderstatisticsaredjSCUssed.
2Preliminaries
Lemma1LetX1,X2,…,Xbei.i.d.
withcdfF(z)andpdff(z),andX1X2…,
X;bethecorrespondingorderstatistics.Then,
thejointdensityfunctionoforderstatisticsisgiven
by
厂1,2,…?(z1,z2,…,X)一,z!f(z),
一
(2o<1?2?…?z<?.(3)
andtheunivariateX…(14s?)marginaldistribu—
tionsof(3)hasdensityfunction,
(z)一EF(x)-IE1一F()]-厂(z),
一
?<<oo.(4)
Wesaythat口(F),definedas
口(F)一inf{:F(z)>0),(5)
isthelowerendpointofthedistributionfunction
F(z).Similarly,theupperendpointco(F)ofF()
isdefinedby
(F)一sup{lz:F(z)<1}.(6)
Lemma2[引LetX1,X2,…,Xbei.i.d.
withthecdfF(z),andX1:,X2:,…,Xbethe
correspondingorderstatistics.Assumethat,for
somefinitea,
r(F)
I(1一F(y))dy<...(7)
Ja
Fora(F)<,<(F),define
r(F)
R(,)一(1一F(,))I(1一F())d.(8)Jt
Assumethat,forallrealz,as(,J(F),
lim一e-or.(9)?n—?_一一?L
Thentherearesequencesaandb>0suchthat,
limP(X…?a+b.x)一H_3IO(z),
where
H3.o(z)一exp(一e一),一oo<x<o~.(10)
Thenormalizingconstantsnandbcanbechosen
n一inf{x:l--F(z)?),…)
and
b一R(a).(12)
Lemma3[]LetXl,X2,…,Xbei.i.d.
withthecdfF(z),andX1X2…,Xbethe
correspondingorderstatistics,anda(F)befinite.
AssumethatthedistributionfunctionF(z)一
F(a(F)一1/x),z<0,satisfiesthefollowingcon—
dition:thereisaconstanty>0suchthat,forallz
>0,asf一?,
lim.)m下一z?
Thentherearesequencescandd>0such
that.as?.
limP(Xl:?c+z)一L2,(1z),
where
z)一
1--exp{--x
..
~…>O.(
14)
Thenormalizingconstantscanddcanbechoose
asc一a(F)and
do=sup{z:F(z)?)一a(F).(15)
3TheMainResults
Theorem1LetXi,X2,…,Xbei.i.d.with
thecdfF(z)in(2),andX,X2:,…,X…bethe
correspondingorderstatistics.Then,forapositive
integerq,wehave
E(Xql)一二T×
?
\一
一?.
一一
一,.
(一h.u
一
1一
一.
S一
?
?
第2期
KUANGNeng—hui:Onpropertiesoforderstatisticsfromthemixedexp0nenti
aldistribun.”137
c硼c,(+
1?s?n,
where=1(1+n3+1)+A2(竹2+竹4),t32一1(n1
+3)+2(2+4+1),andr(z)isGammafunc—
ti.ndefinedby?)一』t~--1e-e
ProofBy(1),(2)and(4),wehave
E(x)一=_x
IzI-F(x)].r[1一F(z)]一厂(z)dx一
二j.xq[1一pe_一(1一)e『]×
[e+(1一)e-A]×
[似e-+(1一).=【2e-x2x]dx—
荛)×
!(二::二e_(AInI+.}×Z16)nowholds.Theproofiscompleted.
Theoreili2LetX1,X2bei.i.d.withthecdf
F(z)in(2),andX1:2,X2:2bethecorresponding
orderstatistics.Then,thesampleintervalsofor—
derstatisticsX1_2,X2:2一X1:2arenotindependent
andnotidenticallydistributed.
ProofLetYl—X1:2,Y2一X2:2一X1;2,andlet
1一1,Y2:z2一X1.Thenzl—yl,z2:==yl+y2,
andtheJacobideterminantJ一1,thus,by(1)and
(3),thejointdensityfunctionof(Y1,y2)isgiven
by
g(Y】,Y2)一f1.2:2(,Yl+yz)lI一
(2[pA1e+(1一p)a2e一z]Epa1e一’+
l(1一)2e2’l2],0?Yl?yl+y2;
Il0,otherwise.
(18)
Therefore,thepdfofY1isgivenby
?
g1(1)===Ig(Yl,y2)d2一
[2Epe-+(1一p)e-zz][纵e
{(1一)2e-2],yl?0;
10,otherwise.
ThepdfofY2isgivenby
gz(y2)一
{2Epa1e+(卜ze][
1(1一)2e.])dyl,y2?0;
l0,otherwise
(19)
pZalea,x,+e+
一
{j掣.+(1一)z.一,?o;l1+A2
【o..therwise.
(2O)
C1early,g(Y1,Y2)?g1(】)g2(2),thustheproof
isfinished.
Similarly,weobtain
TheOreii13LetX1,X2,…,Xbei.i.d.with
thecdfF()in(2),andX1:,X2:,…,X…bethe
correspondingorderstatistics.Then,thesample
interva1soforderstatisticsX1:,X2:一X1:,…,
X一X一1arenotindependentandnotidentical—
lydistributed.
Theorenl4LetX1,X2,…,Xbei.i.d.with
thecdfF(z)in(2),andX1:,X2:,…,X:bethe
correspondingorderstatistics.Thentherearese—
queneesaandb>0suchthat,as,z?,
limP(X…?口+635)=::H(z),
where
H圳(z)一exp(--e一),一?<z<CXD.(21)
Thenormalizingconstantsaandbcanbe
ll
y
d
)
2
/L
g
??
+
???J
(一卜毫
,,??,一
1一?.一.一m
罄
138浙江大学(理学版)第38卷
chosenas
n一
叫z:1一F(z)?)(22)
and
b=R(口),(23)
where
R(,)一(1一F())l(1一F())dy.(24)
ProofForthecdfF()in(2),onecaneasi-
lYcheckthat?(F)一?.Since
r..r..l(1一F())d—I[e一-+(1一)e-~2s]d.y—Jn?n
en+en.(25)
A2
Therefore(7)evidentlyholdswith,say,n—1.
Thefollowingstepistocheck(9).Notethat,ast
??,byL’Hospital’srule,
limR(f)一1im(1一F())
im1二peal
ProofForthecdfF(x)in(2),onecaneasily
checkthata(F)一0.Hence,F()===F(一1/x),z<
0,andas,十一?,byL’Hospital’srule,
F(一1)
F(一?)
1一ea1一
(1一户)
e
e1一pe一(1一户)
lim
Z
pA1
1
e”
e
+(1一p)A2e7
P)2eT
applies
1
Z
withy
(30)
The
proofiscompleted.
Itshouldalsobeemphasizedthatthecon
stantscanddarenotunique.
(1一F())dy—References:
?二::=
+(1一p)g2e一.[131e,l
?二2呈二二
户1+(1一p)X2e-(a2一l一_
1
,(26)
A1
therefore,weobtain,forallrea1272,as,?,
..1一F(t+xR())n
m—?一一
()?二竺
+(1一P)e
(tt)
pe一lA2t
+(1一.(
P+(1一户)e1%-al一e.
(27)
Thus(9)holds.ByLemma2,theproofiscomple—
ted.
Itshouldbeemphasizedthattheconstantsa
andbarenotunique.
Theorem5LetXl,X2,…,Xbei.i.d.with
thecdfF(x)in(2),andXlX2_”,…,Xbethe
correspondingorderstatistics.Thentherearese—
quencescandd>0suchthat,as72?,
limP(X1:?c+d32)一L2,y(),
where
L2,r
(x)一{.,i1--expz{--<x}”?.’
andthenormalizingconstantscanddcanbe
chooseasc一0and
一sup
{z:F()?).(29)
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第2期
KUANGNeng—hui:Onpropertiesoforderstatisticsfromthemixedexponentialdistribution139
(上接第125页)
乎可裂正合列.
证明由命题2,有(XU).是不可分解非投
射kG一模当且仅当(y).是不可分解非投射kL一
模,结论直接由定理1和4得到.
推论1设x是不可分解非投射kG一模,y是
相应的不可分解非投射志L一模.则PtdimX当且仅
当PtdimY.
证明因为k是代数闭域,直接由定理1和2
得到结论.
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