【word】 混合指数分布顺序统计量的性质

【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) [2] [3] [4] [5] [6] [7] [8] [9] rIANYZ.WANGBC,CHENP.Parametersestl— mationforamixtureofgeneralizedexponentialdistri— butions[J].JiangxiNormalUniversity:NaturalSci- ence,2009,33(3):297—300. GUPTARD.KUNDUD.Generalizedexponential distribution:Bayesianestimation[J].StatistDataA— nal,2008,52:1873—1883. IMREC.FRANTISEKM.Generalizedmaximum likelihoodestimatesforexponentialfamilies[J].Prob— abilityTheoryandRelatedFields,2008,141:213—246. YINLP.WANGNH,LUYN,eta1.Aclassofre— newalriskprocessinwhichtheclaimintervalisunder themixedexponentialdistribution[J].Mathematicsin Economics,2008,25(2):ll8,125. PANDEYM.UPADHYAYSK.Bayesianinference inmixturesoftwoexponentials[J].Microelectronics Reliability,1988,28(2):217—221. JAROSLAWB.Mixturesofexponentialdistributions andstochasticorders[J].StatisticsandProbabilityLet- ters,2002,57(1):23—31. GUPTARD,KUNDUD.Generalizedexponential distribution:existingresultsandsomerecentdevelop— ments[J].StatistPlanInf,2007,104:339—350. GIUIlANAR.Aclassofbivariateexponentialdistri— butions[J].JMultivariateAnalysis,2009,100(6):1261 — 1269. KUNDUD,GUPTARD.Bivariategeneralizedexpo— nentialdistr|bution[J].JMultivariateAnalysis,2009, 1OO(4):581—593. 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. 参考文献(References): El-1AUSLANDERM,REITENI,SMALOSO.Repre- sentationTheoryofArtinAlgebras[M].Cambridge: CambridgeUniversityPress,1997. Ez]GARLINGD,GORENSTEIND,DIECKTTom,etal. LocalRepresentationTheory:CambridgeStudiesinAd— vancesMathematicsII[M].Cambridge:CambridgeUni— versityPress,1993.(责任编辑寿彩丽)