基于置信分布的多个对数正态总体的公共均值的置信区间的构建（英文）-豆柴文库

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基于置信分布的多个对数正态总体的公共均值的置信区间的构建（英文）
ConfidenceIntervalConstructionfortheCommonMeanofMultipleLognormalPopulationsBasedonConfidenceDistribution
Introduction:
Inmanysituations,itisessentialtoestimateandconstructconfidenceintervalsforthecommonmeanofmultiplepopulations.Onecommonapproachistousethemethodsbasedonthenormaldistribution,assumingthatthepopulationsarenormallydistributed.However,inmanycases,thedatadonotfollownormaldistributions,andonepotentialalternativeistousethelognormaldistribution,whichisoftenobservedinreal-lifedata.Inthispaper,wepresentamethodforconstructingtheconfidenceintervalofthecommonmeanofmultiplelognormalpopulationsbasedontheconfidencedistribution.
Methodology:
Supposewehaveklognormalpopulations,andletYijbethejthobservationfromtheithpopulation,wherei=1,2,...,k;j=1,2,...,ni.Lettherandomvariablesln(Yij)followthenormaldistributionwithunknownmeanμiandvarianceσ^2.Assumingthatallthepopulationshavethesamevarianceσ^2,wecanusethepooledestimatorofthevariance:
S^2=(1/(n1+n2+...+nk-k))*[(n1-1)S1^2+(n2-1)S2^2+...+(nk-1)Sk^2]
whereS1^2,S2^2,...,Sk^2arethesamplevariancesofthekpopulations.Thentheestimatorofthecommonmeanμis:
M=(1/k)*[(n1/Y1)+(n2/Y2)+...+(nk/Yk)]
whereY1,Y2,...,Ykarethesamplemeansofthekpopulations.
Theconfidencedistributionisthenusedtoconstructtheconfidenceintervalofthecommonmeanμwithagivenconfidencelevel1-α.Theconfidencedistributionisusedtoestimatetheconfidencelevelofarandomintervalwhentheunderlyingdistributionisnotknown.WefirsttransformtherandomvariableMintoastandardnormaldistributionbyusingthetransformation:
Z=(M-μ)/(S/√k)
whereSistheestimatorofthestandarddeviationoftheestimatorM,whichisS=S/√k.
Thenwecanconstructtheconfidencedistributionfunctionbasedonthetransformeddata,denotedasF(Z;D).Thelevel1-αconfidenceintervalofthecommonmeanμisthenconstructedbyfindingthetwopointsaandbsuchthat:
P(a≤Z≤b;D)=F(b;D)-F(a;D)=1-α
whereDdenotestheprobabilitydistributionoftherandomvariableZ.Theinterval[μ+aS/√k,μ+bS/√k]istheconfidenceintervalofthecommonmeanμ.
Results:
Toeval