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1、QuantitativeMethodsinFinance:TimeSeriesAnalysisMarcS.PaolellaMarch2005ParameterEstimationandLikelihoodDataisnotinformation,informationisnotknowledge,knowledgeisnotunderstanding,understandingisnotwis-dom.(Cli?StollandGarySchubert)Donotputfaithinwhatstatisticssayuntilyouhavecare-fullyconsidere
2、dwhattheydonotsay.(WilliamW.Watt)Commonsense"isnotcommonbutneedstolearntsys-tematically....Asimpleanalysis"canbeharderthanitlooks....Allstatisticaltechniques,howeversophisticated,shouldbesubordinatetosubjectivejudgement.(Chatˉeld,1985,p.228)12Question:Interestcentersonhowlongittakesawomant
3、obecomepregnantusingaparticularmethodofassis-tance,e.g.,temperaturemeasurements,hormonetreat-ment,etc.Imagineastudyinwhicheachofryoungcouples(independentlyfromeachother)attemptstocon-ceiveeachmonthuntiltheysucceed".Lettherequirednumberofmonthsforeachcouplebede-notedbyXi,i=1;:::;r.Givenvalue
4、sX1=x1;:::;Xr=xr,howwouldyoucomputePr(X>5),i.e.,theproba-bilitythatittakesmorethan5monthstoconceive?AnswerI:Ingeneral,giventheindependentandidenticallydis-tributed(iid)datasetX1;:::;Xn,considertheratherintuitiveestimatorofPr(X·t):Xn?1pbt=Pr(X·t)=nI(?1;t)(Xi)i=1whereI(?1;t)(Xi)istheindicatorf
5、unction,deˉnedforeverysetMμRas(1;ifx2M;IM(x)=0;ifx=2M:Thisiscalledtheempiricalcumulativedistributionfunc-tion,abbreviatedtheempiricalcdf,orjusttheecdf.3Let'ssimulateasetof15couplesusingMatlab:p=0.3;%probabilityofsuccessinanygivenmonthn=15;%howmanycouplesdata=geornd(0.3,15,1)+1%geometricrando
6、mvariables!Herearesometypicalvalues:6131542438123243Nowweneedtocomputeandplottheempiricalcdf.Firsttabulatetheresultswitht=tabulate(data),giving1213.332320.003320.004320.00516.66616.6670816.66901001101201316.664Nowlet'scomputetheecdfandplotit:ecdf=cumsum(t(:,2));grd=t(:,1);plot(grd,ecdf/n,'b-
7、o'),set(gca,'fontsize',14)h=line([55],[01]);set(h,'linestyle','--')h=line([05],[0.80.8]);set(h,'linestyle',':','linewidth',2)10.90.80.70.60.50.40.30.20.1002468101214Figure1:Empiricalcdf2Nowwecananswerouroriginalquestion:Fromtheecdf,weseethatanestim