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20084222JournalofAcademyofArmoredForceEngineeringApr.2008Vo.l22No.2:16721497(2008)02008803潘高田党明涛王保恒赵东波(,100072):n(0,1)XP(n),Y=-lnXGamma(n,1),:;;;:O2121:AResearchesontheStatisticofSomeSmallSampletestsPANGaotianDANGMingtaoWANGBaohengZHAODongbo(DepartmentofEquipmentCommandandAdministration,AcademyofArmoredForceEngineering,Beijing100072,China)Abstract:LetXbetheproductofnindependentandrandomvariableseachwiththesamedistributionastheuniformdistributionwithparameters0and1.TherandomvariableXfollowsthePdistributionwithndegreesoffreedomdefinedinthispaper.ItsprovedthatthenewrandomvariableY=-lnXconformstoGammadistributionwithparametersnand1.Accordingtothisconclusion,thepapermainlyclarifiestheoriesandmethodsofsmallsampleparametertestaboutnon-normaldistribution,discussesthesmallsampletestproblemofexponentialdistribution,Weibulldistributionand!∀#!distribution.Keywords:exponentdistribution;Weibulldistribution;!∀#!distribution;hypothesistest:20071126:(1955-),,,,.,,(!∀#!),,1[1],XF(x),Y=F(X),Y(0,1),Y~U(0,1)U(0,1)n(Y1,Y2,Yn),P(n)=Y1Y2Yn,P(n)1:P(n)FP(n)(Z)=0,z0z1+!ni=2(-lnz)i-1(i-1)!,0∀z11,z#1.(1)P(n)fP(n)(z)=(-1)n-1(n-1)!(lnz)n-1,0z10,.(2):P(n)(2),(2)(1)k=1,P(1)=Y1,f(z)=1,0z10,,2:k=n,,fP(n)(z)=(-lnz)n-1(n-1)!,0z10,.k=n+1,P(n+1)=Y1Y2YnYn+1,P(n+1)=P(n)Yn+1,Yn+1(0,1),P(n)Yn+1,P(n+1)FP(n+1)(z)=P{P(n)Yn+1∀z}=xy∀z0x,y1(-lny)n-1(n-1)!dxdy.z0,FP(n+1)(z)=0;z#1,FP(n+1)(z)=1;0∀z1,FP(n+1)(z)=1-∃1z(-lny)n-1(n-1)!dy∃1zydx=1-∃1z(-lny)n-1(n-1)!dy-∃1z(-lny)n-1(n-1)!zydy,∃1z(-lny)n-1(n-1)!zydy=(-1)n-1(n-1)!z∃1z(lny)n-1d(lny)=(-1)n-1n!z(lny)n1z=(-1)nn!z(lnz)n.,FP(n+1)(z)=1-∃1z(-lny)n-1(n-1)!dy-(-1)nn!z(lnz)n.,:fP(n+1)(z)=--(-lnz)n-1(n-1)!-(-1)nn!ddz[z(lnz)n]=(-1)nn!(lnz)n,2:XP(n),Y=-lnX,YGamma∃(n,1):FY(y)=P{Y∀y}=P{-lnX∀y}=P{X#e-y}=∃1e-y(-lnx)n-1(n-1)!dx,y00,y∀0,fY(y)=FY(y)=yn-1(n-1)!e-y,y00,y∀0.Gamma∃(n,1)1222.1X:f(x)=1%exp(-x%),x00,x∀0,F(x)=1-exp(-x%),x00,x∀0.X1,X2,,Xn,X,Ui=1-exp(-Xi%),i=1,2,,n,,Ui(i=1,2,,n)(0,1)1-UiU(0,1),,Vi=exp(-Xi%)U(0,1):p=V1V2Vn=%ni=1exp(-Xi%)=exp-!ni=1Xi%=exp-nX%.1p~P(n),2nX&%Gamma∃(n,1),H0∋%#(∀,=)%0,H1∋%(,()%0,nX&%02.2(!∀#!)()()T,!∀#!F(t)=1-1+2tt0e-2tt0,t#00,t0,t0;F(t),E(T)=t0t0T1,T2,,Tn,Ui=1-(1+2Tit0)e-2Tit0,i=1,2,,n,21:8922p=V1V2Vn=%ni=11+2Tit0e-2Tit0=e-2!ni=1Tit0%ni=11+2Tit0,Y=-lnp=2!ni=1Tit0-!ni=1ln1+2Tit0.(3)2,YGamma∃(n,1),H0∋t0#(∀,=)t~0,H1∋t0(,()t~0,(3)2.3T,,,F(t)=1-e-(ta)b,t#00,t0.H0∋b=b0,a=a0;b=b0,a=a0;a=a0,b=b0P(n)b=b0,a=a0a0H0∋a#(∀,=)a0,H1∋a(,()a0,T1,T2,,Tn,2.1P(n)=%ni=1e-(tia0)b0=e-!ni=1(tia0)b02,!ni=1(tia0)b0Gamma∃(n,1),!ni=1(tia0)b03X,&,H0∋%#%0,H1∋%%0,nX&%0Y=nX&%0%0,,nX&%∀nX&%0,{nX&%k}{nX&%0k},P{nX&%0∃1-&(n,1)}∀P{nX&%∃1-&(n,1)}=&,W={(X1,X2,,Xn)nX&%0∃1-&(n,1)}.,H0∋%∀%0,H1∋%%0W={(X1,X2,,Xn)nX&%0∃&(n,1)};H0∋%=%0,H1∋%(%0W={(X1,X2,,Xn)nX&%0∃1-&2(n,1)nX&%0∃&2(n,1)}.,P{∃1-&2(n,1)nX&%0∃&2(n,1)}=1-&,,%01-&(nX&∃&2(n,1),nX&∃1-&2(n,1)).1:,10,:05045,16155,02920,00706,08708,13012,41799,08095,07636,01670.!10i=1Xi=105746,&=005,∃&(10,1)=157052,∃1-&(10,1)=54254,∃&2(10,1)=170848,∃1-&2(10,1)=47954,%095%(06319,22052):[1],.[M].:,2007:96-97.[2],,.[M].:,1998.[3].[M].:,1986.[4].[M].:,1995.[5],.[M].:,1993.[6].[M].:,1999.(:)90