diff --git a/.gitignore b/.gitignore index 44442b1..0a6c9d4 100644 --- a/.gitignore +++ b/.gitignore @@ -27,5 +27,6 @@ *.mtc *.mtc1 *.out +*.pdf *.synctex.gz /GAL-Buckle95-concordance.tex diff --git a/GAL-Buckle95.Rnw b/GAL-Buckle95.Rnw index f98ed3e..27b407a 100644 --- a/GAL-Buckle95.Rnw +++ b/GAL-Buckle95.Rnw @@ -19,6 +19,8 @@ \maketitle \tableofcontents +\chapter{Initialisation} + \section{Chargement des paquets} <<>>= setwd("~/git/GAL-Buckle95/") @@ -56,17 +58,19 @@ n <- length(RETURNS) EppsPulley.test(RETURNS) @ +\chapter{Estimation} + \section{Données mises à l'échelle} <<>>= -scaledRETURNS <- as.vector(scale(RETURNS)) +sRET <- as.vector(scale(RETURNS)) @ \section{Première estimation par QEE} <<>>= ## Point de départ -pt.depart <- startparamGAL(scaledRETURNS) +pt.depart <- startparamGAL(sRET) ## Fonctions pour les moments meanQEE <- function(param) mGAL(param,1) varianceQEE <- function(param) cmGAL(param,2) @@ -77,72 +81,197 @@ kurtosisQEE <- function(param) cmGAL(param,4) dmeanQEE <- function(param) dmGAL(param,1) dsdQEE <- function(param) dmGAL(param,2) ## Estimation gaussienne -optim1 <- optim(pt.depart,obj.gauss,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE) +optim1 <- optim(pt.depart,obj.gauss,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE) pt.optim1 <- optim1$par ## Estimation de crowder -optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE) +optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,sRET, + meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE) pt.optim2 <- optim2$par ## Estimation de crowder modifiée -optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE) +optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE) pt.optim3 <- optim3$par @ \section{Résultats de la première estimation par QEE} <<>>= - cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE), - V.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) - cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), - V.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) - cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), - V.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE),n) - confidence.interval.QEE(pt.optim1,cov.optim1,n) - confidence.interval.QEE(pt.optim2,cov.optim2,n) - confidence.interval.QEE(pt.optim3,cov.optim3,n) +cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE), + V.gauss(pt.optim1,sRET,meanQEE,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder(pt.optim2,sRET,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder.Mod(pt.optim3,sRET,varianceQEE,dmeanQEE,dsdQEE),n) +confidence.interval.QEE(pt.optim1,cov.optim1,n) +confidence.interval.QEE(pt.optim2,cov.optim2,n) +confidence.interval.QEE(pt.optim3,cov.optim3,n) @ \section{Seconde estimation par QEE} <<>>= ## Estimation gaussienne -optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE, - ginv(V.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE))) +optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE, + ginv(V.gauss(pt.optim1,sRET,meanQEE, + varianceQEE,skewnessQEE,kurtosisQEE, + dmeanQEE,dsdQEE))) pt.optim4 <- optim4$par ## Estimation de crowder -optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE, - ginv(V.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE))) +optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,sRET, + meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE, + ginv(V.Crowder(pt.optim2,sRET,varianceQEE,skewnessQEE, + kurtosisQEE,dmeanQEE,dsdQEE))) pt.optim5 <- optim5$par ## Estimation de crowder modifiée -optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE, - ginv(V.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE))) +optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE, + ginv(V.Crowder.Mod(pt.optim3,sRET,varianceQEE, + dmeanQEE,dsdQEE))) pt.optim6 <- optim6$par @ \section{Résultats de la seconde estimation par QEE} <<>>= - cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE), - V.gauss(pt.optim4,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) - cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), - V.Crowder(pt.optim5,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) - cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), - V.Crowder.Mod(pt.optim6,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE),n) - confidence.interval.QEE(pt.optim4,cov.optim4,n) - confidence.interval.QEE(pt.optim5,cov.optim5,n) - confidence.interval.QEE(pt.optim6,cov.optim6,n) +cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,sRET, + meanQEE,varianceQEE,dmeanQEE,dsdQEE), + V.gauss(pt.optim4,sRET,meanQEE,varianceQEE, + skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder(pt.optim5,sRET,varianceQEE,skewnessQEE, + kurtosisQEE,dmeanQEE,dsdQEE),n) +cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,sRET, + varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), + V.Crowder.Mod(pt.optim6,sRET,varianceQEE,dmeanQEE,dsdQEE),n) +confidence.interval.QEE(pt.optim4,cov.optim4,n) +confidence.interval.QEE(pt.optim5,cov.optim5,n) +confidence.interval.QEE(pt.optim6,cov.optim6,n) @ \section{Estimation par GMM} <<>>= - ## GMM régulier - optim7 <- optim.GMM(pt.depart,conditions.vector=meanvariance.gmm.vector,data=scaledRETURNS,W=diag(2), - meanf=meanQEE,variancef=varianceQEE) - - ## GMM itératif - optim8 <- iterative.GMM(pt.depart,conditions.vector=meanvariance.gmm.vector,data=scaledRETURNS,W=diag(2), - meanf=meanQEE,variancef=varianceQEE) +## GMM régulier +optim7 <- optim.GMM(pt.depart, + conditions.vector=meanvariance.gmm.vector, + data=sRET,W=diag(2), + meanf=meanQEE,variancef=varianceQEE) +pt.optim7 <- optim7$par +cov.optim7 <- mean.variance.GMM.gradient.GAL(pt.optim7,sRET) %*% + covariance.GMM(meanvariance.gmm.vector,pt.optim7,sRET, + meanf=meanQEE,variancef=varianceQEE) %*% + t(mean.variance.GMM.gradient.GAL(pt.optim7,sRET)) / n +## GMM itératif +optim8 <- iterative.GMM(pt.depart, + conditions.vector=meanvariance.gmm.vector, + data=sRET,W=diag(2), + meanf=meanQEE,variancef=varianceQEE) +pt.optim8 <- optim8$par +cov.optim8 <- mean.variance.GMM.gradient.GAL(pt.optim8,sRET) %*% + optim8$cov %*% + t(mean.variance.GMM.gradient.GAL(pt.optim8,sRET)) / n +confidence.interval.QEE(pt.optim7,cov.optim7,n) +confidence.interval.QEE(pt.optim8,cov.optim8,n) +@ + +\chapter{Comparaison des résultats} +<<>>= +# Aggrégation des estimateurs (pour simplifier les calculs) +pts.estim <- cbind(pt.optim1,pt.optim2,pt.optim3,pt.optim4, + pt.optim5,pt.optim6,pt.optim7,pt.optim8) +l.pts.estim <- as.list(data.frame(pts.estim)) +@ + +\section{Fonction de répartition} +<<>>= +# Points d'évaluation +xi <- seq(2*min(sRET),2*max(sRET),length.out=2^6) +# Fonction de répartition par intégration de la fonction caractéristique +dist1 <- cbind(cftocdf(xi,cfGAL,param=pt.optim1), + cftocdf(xi,cfGAL,param=pt.optim2), + cftocdf(xi,cfGAL,param=pt.optim3), + cftocdf(xi,cfGAL,param=pt.optim4), + cftocdf(xi,cfGAL,param=pt.optim5), + cftocdf(xi,cfGAL,param=pt.optim6), + cftocdf(xi,cfGAL,param=pt.optim7), + cftocdf(xi,cfGAL,param=pt.optim8)) +# Fonction de répartition par point de selle +dist2 <- cbind(psaddleapproxGAL(xi,pt.optim1), + psaddleapproxGAL(xi,pt.optim2), + psaddleapproxGAL(xi,pt.optim3), + psaddleapproxGAL(xi,pt.optim4), + psaddleapproxGAL(xi,pt.optim5), + psaddleapproxGAL(xi,pt.optim6), + psaddleapproxGAL(xi,pt.optim7), + psaddleapproxGAL(xi,pt.optim8)) +# Fonction de répartition par intégration de la fonction de densité +dist3 <- cbind(pGAL(xi,pt.optim1), + pGAL(xi,pt.optim2), + pGAL(xi,pt.optim3), + pGAL(xi,pt.optim4), + pGAL(xi,pt.optim5), + pGAL(xi,pt.optim6), + pGAL(xi,pt.optim7), + pGAL(xi,pt.optim8)) +@ +\pagebreak +\subsection{Graphiques} + +<>= + for (i in 1:8) + { + file<-paste("dist-GAL-",i,".pdf",sep="") + pdf(file=file,paper="special",width=6,height=6) + plot.ecdf(sRET,main=paste("Fonction de répartition ",i)) + lines(xi,dist1[,i],col="green") + lines(xi,dist2[,1],col="red") + lines(xi,dist3[,1],col="pink") + lines(xi,pnorm(xi),type="l",col="blue") + dev.off() + cat("\\includegraphics[height=2in,width=2in]{", + file,"}\n",sep="") + } +@ + +\subsection{Statistiques} +Test du $\chi^2$, Méthode avec intégration +<>= +chisquare.test1 <- function(param,DATA.hist,FUN,method) +{ + chisquare.test(DATA.hist,FUN,param,method=method) +} +xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),cfGAL,"integral")),digits=6) +@ + +Test du $\chi^2$, Méthode avec point de selle +<>= +xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),pGAL,"saddlepoint")),digits=6) +@ + +Statistique de Kolmogorov-Smirnov +<>= + ks.test1 <- function(param,x,y) ks.test(x,y,param) + xtable(do.call(rbind,mclapply(l.pts.estim,ks.test1,sRET,"pGAL")),digits=6) +@ + +Statistique de distance minimale + +<>= + tvariate1 <- seq(-.1,.1,by=0.01) + xtable(do.call(rbind,mclapply(l.pts.estim,md.test,sRET,tvariate1,cfGAL,empCF)),digits=6) @ + + + + \end{document} diff --git a/GAL-Buckle95.pdf b/GAL-Buckle95.pdf index a8f63eb..ff00ca9 100644 Binary files a/GAL-Buckle95.pdf and b/GAL-Buckle95.pdf differ diff --git a/GAL-Buckle95.tex b/GAL-Buckle95.tex index 2bfa65e..4d30be4 100644 --- a/GAL-Buckle95.tex +++ b/GAL-Buckle95.tex @@ -19,6 +19,8 @@ \maketitle \tableofcontents +\chapter{Initialisation} + \section{Chargement des paquets} \begin{Schunk} \begin{Sinput} @@ -84,11 +86,13 @@ $Reject \end{Soutput} \end{Schunk} +\chapter{Estimation} + \section{Données mises à l'échelle} \begin{Schunk} \begin{Sinput} -> scaledRETURNS <- as.vector(scale(RETURNS)) +> sRET <- as.vector(scale(RETURNS)) \end{Sinput} \end{Schunk} @@ -97,7 +101,7 @@ $Reject \begin{Schunk} \begin{Sinput} > ## Point de départ -> pt.depart <- startparamGAL(scaledRETURNS) +> pt.depart <- startparamGAL(sRET) > ## Fonctions pour les moments > meanQEE <- function(param) mGAL(param,1) > varianceQEE <- function(param) cmGAL(param,2) @@ -108,13 +112,16 @@ $Reject > dmeanQEE <- function(param) dmGAL(param,1) > dsdQEE <- function(param) dmGAL(param,2) > ## Estimation gaussienne -> optim1 <- optim(pt.depart,obj.gauss,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE) +> optim1 <- optim(pt.depart,obj.gauss,gr=NULL,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE) > pt.optim1 <- optim1$par > ## Estimation de crowder -> optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE) +> optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,sRET, ++ meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE) > pt.optim2 <- optim2$par > ## Estimation de crowder modifiée -> optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE) +> optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE) > pt.optim3 <- optim3$par \end{Sinput} \end{Schunk} @@ -123,13 +130,18 @@ $Reject \begin{Schunk} \begin{Sinput} -> cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE), -+ V.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) -> cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), -+ V.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) -> cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), -+ V.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE),n) -> confidence.interval.QEE(pt.optim1,cov.optim1,n) +> cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE), ++ V.gauss(pt.optim1,sRET,meanQEE,varianceQEE, ++ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +> cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,sRET, ++ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), ++ V.Crowder(pt.optim2,sRET,varianceQEE, ++ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +> cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,sRET, ++ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), ++ V.Crowder.Mod(pt.optim3,sRET,varianceQEE,dmeanQEE,dsdQEE),n) +> confidence.interval.QEE(pt.optim1,cov.optim1,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -139,7 +151,7 @@ $Reject [4,] 1.994757 2.021370 2.047982 \end{Soutput} \begin{Sinput} -> confidence.interval.QEE(pt.optim2,cov.optim2,n) +> confidence.interval.QEE(pt.optim2,cov.optim2,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -149,7 +161,7 @@ $Reject [4,] 1.839966 1.878296 1.916626 \end{Soutput} \begin{Sinput} -> confidence.interval.QEE(pt.optim3,cov.optim3,n) +> confidence.interval.QEE(pt.optim3,cov.optim3,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -165,16 +177,23 @@ $Reject \begin{Schunk} \begin{Sinput} > ## Estimation gaussienne -> optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE, -+ ginv(V.gauss(pt.optim1,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE))) +> optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE, ++ ginv(V.gauss(pt.optim1,sRET,meanQEE, ++ varianceQEE,skewnessQEE,kurtosisQEE, ++ dmeanQEE,dsdQEE))) > pt.optim4 <- optim4$par > ## Estimation de crowder -> optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE, -+ ginv(V.Crowder(pt.optim2,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE))) +> optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,sRET, ++ meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE, ++ ginv(V.Crowder(pt.optim2,sRET,varianceQEE,skewnessQEE, ++ kurtosisQEE,dmeanQEE,dsdQEE))) > pt.optim5 <- optim5$par > ## Estimation de crowder modifiée -> optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE, -+ ginv(V.Crowder.Mod(pt.optim3,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE))) +> optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE, ++ ginv(V.Crowder.Mod(pt.optim3,sRET,varianceQEE, ++ dmeanQEE,dsdQEE))) > pt.optim6 <- optim6$par \end{Sinput} \end{Schunk} @@ -183,13 +202,18 @@ $Reject \begin{Schunk} \begin{Sinput} -> cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,scaledRETURNS,meanQEE,varianceQEE,dmeanQEE,dsdQEE), -+ V.gauss(pt.optim4,scaledRETURNS,meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) -> cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), -+ V.Crowder(pt.optim5,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) -> cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,scaledRETURNS,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), -+ V.Crowder.Mod(pt.optim6,scaledRETURNS,varianceQEE,dmeanQEE,dsdQEE),n) -> confidence.interval.QEE(pt.optim4,cov.optim4,n) +> cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,sRET, ++ meanQEE,varianceQEE,dmeanQEE,dsdQEE), ++ V.gauss(pt.optim4,sRET,meanQEE,varianceQEE, ++ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n) +> cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,sRET, ++ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), ++ V.Crowder(pt.optim5,sRET,varianceQEE,skewnessQEE, ++ kurtosisQEE,dmeanQEE,dsdQEE),n) +> cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,sRET, ++ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE), ++ V.Crowder.Mod(pt.optim6,sRET,varianceQEE,dmeanQEE,dsdQEE),n) +> confidence.interval.QEE(pt.optim4,cov.optim4,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -199,7 +223,7 @@ $Reject [4,] 1.995452 2.022048 2.048644 \end{Soutput} \begin{Sinput} -> confidence.interval.QEE(pt.optim5,cov.optim5,n) +> confidence.interval.QEE(pt.optim5,cov.optim5,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -209,7 +233,7 @@ $Reject [4,] 1.842116 1.880376 1.918636 \end{Soutput} \begin{Sinput} -> confidence.interval.QEE(pt.optim6,cov.optim6,n) +> confidence.interval.QEE(pt.optim6,cov.optim6,n) \end{Sinput} \begin{Soutput} LOWER ESTIMATE UPPER @@ -224,14 +248,229 @@ $Reject \begin{Schunk} \begin{Sinput} -> ## GMM régulier -> optim7 <- optim.GMM(pt.depart,conditions.vector=meanvariance.gmm.vector,data=scaledRETURNS,W=diag(2), -+ meanf=meanQEE,variancef=varianceQEE) -> ## GMM itératif -> optim8 <- iterative.GMM(pt.depart,conditions.vector=meanvariance.gmm.vector,data=scaledRETURNS,W=diag(2), -+ meanf=meanQEE,variancef=varianceQEE) +> ## GMM régulier +> optim7 <- optim.GMM(pt.depart, ++ conditions.vector=meanvariance.gmm.vector, ++ data=sRET,W=diag(2), ++ meanf=meanQEE,variancef=varianceQEE) +> pt.optim7 <- optim7$par +> cov.optim7 <- mean.variance.GMM.gradient.GAL(pt.optim7,sRET) %*% ++ covariance.GMM(meanvariance.gmm.vector,pt.optim7,sRET, ++ meanf=meanQEE,variancef=varianceQEE) %*% ++ t(mean.variance.GMM.gradient.GAL(pt.optim7,sRET)) / n +> ## GMM itératif +> optim8 <- iterative.GMM(pt.depart, ++ conditions.vector=meanvariance.gmm.vector, ++ data=sRET,W=diag(2), ++ meanf=meanQEE,variancef=varianceQEE) +> pt.optim8 <- optim8$par +> cov.optim8 <- mean.variance.GMM.gradient.GAL(pt.optim8,sRET) %*% ++ optim8$cov %*% ++ t(mean.variance.GMM.gradient.GAL(pt.optim8,sRET)) / n +> confidence.interval.QEE(pt.optim7,cov.optim7,n) +\end{Sinput} +\begin{Soutput} + LOWER ESTIMATE UPPER +[1,] -0.878702 -0.641646 -0.404589 +[2,] -0.469225 0.625908 1.721040 +[3,] -0.192234 0.326366 0.844965 +[4,] 1.696121 1.965995 2.235869 +\end{Soutput} +\begin{Sinput} +> confidence.interval.QEE(pt.optim8,cov.optim8,n) +\end{Sinput} +\begin{Soutput} + LOWER ESTIMATE UPPER +[1,] -0.874031 -0.636980 -0.399929 +[2,] -0.473292 0.626346 1.725984 +[3,] -0.193600 0.322895 0.839390 +[4,] 1.704166 1.972716 2.241265 +\end{Soutput} +\end{Schunk} + +\chapter{Comparaison des résultats} +\begin{Schunk} +\begin{Sinput} +> # Aggrégation des estimateurs (pour simplifier les calculs) +> pts.estim <- cbind(pt.optim1,pt.optim2,pt.optim3,pt.optim4, ++ pt.optim5,pt.optim6,pt.optim7,pt.optim8) +> l.pts.estim <- as.list(data.frame(pts.estim)) \end{Sinput} \end{Schunk} +\section{Fonction de répartition} +\begin{Schunk} +\begin{Sinput} +> # Points d'évaluation +> xi <- seq(2*min(sRET),2*max(sRET),length.out=2^6) +> # Fonction de répartition par intégration de la fonction caractéristique +> dist1 <- cbind(cftocdf(xi,cfGAL,param=pt.optim1), ++ cftocdf(xi,cfGAL,param=pt.optim2), ++ cftocdf(xi,cfGAL,param=pt.optim3), ++ cftocdf(xi,cfGAL,param=pt.optim4), ++ cftocdf(xi,cfGAL,param=pt.optim5), ++ cftocdf(xi,cfGAL,param=pt.optim6), ++ cftocdf(xi,cfGAL,param=pt.optim7), ++ cftocdf(xi,cfGAL,param=pt.optim8)) +> # Fonction de répartition par point de selle +> dist2 <- cbind(psaddleapproxGAL(xi,pt.optim1), ++ psaddleapproxGAL(xi,pt.optim2), ++ psaddleapproxGAL(xi,pt.optim3), ++ psaddleapproxGAL(xi,pt.optim4), ++ psaddleapproxGAL(xi,pt.optim5), ++ psaddleapproxGAL(xi,pt.optim6), ++ psaddleapproxGAL(xi,pt.optim7), ++ psaddleapproxGAL(xi,pt.optim8)) +> # Fonction de répartition par intégration de la fonction de densité +> dist3 <- cbind(pGAL(xi,pt.optim1), ++ pGAL(xi,pt.optim2), ++ pGAL(xi,pt.optim3), ++ pGAL(xi,pt.optim4), ++ pGAL(xi,pt.optim5), ++ pGAL(xi,pt.optim6), ++ pGAL(xi,pt.optim7), ++ pGAL(xi,pt.optim8)) +\end{Sinput} +\end{Schunk} +\pagebreak +\subsection{Graphiques} + +\begin{Schunk} +\begin{Sinput} +> for (i in 1:8) ++ { ++ file<-paste("dist-GAL-",i,".pdf",sep="") ++ pdf(file=file,paper="special",width=6,height=6) ++ plot.ecdf(sRET,main=paste("Fonction de répartition ",i)) ++ lines(xi,dist1[,i],col="green") ++ lines(xi,dist2[,1],col="red") ++ lines(xi,dist3[,1],col="pink") ++ lines(xi,pnorm(xi),type="l",col="blue") ++ dev.off() ++ cat("\\includegraphics[height=2in,width=2in]{", ++ file,"}\n",sep="") ++ } +\end{Sinput} +\includegraphics[height=2in,width=2in]{dist-GAL-1.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-2.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-3.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-4.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-5.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-6.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-7.pdf} +\includegraphics[height=2in,width=2in]{dist-GAL-8.pdf}\end{Schunk} + +\subsection{Statistiques} +Test du $\chi^2$, Méthode avec intégration +\begin{Schunk} +\begin{Sinput} +> chisquare.test1 <- function(param,DATA.hist,FUN,method) ++ { ++ chisquare.test(DATA.hist,FUN,param,method=method) ++ } +> xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),cfGAL,"integral")),digits=6) +\end{Sinput} +% latex table generated in R 3.0.2 by xtable 1.7-1 package +% Sat Mar 15 11:38:58 2014 +\begin{table}[ht] +\centering +\begin{tabular}{rrrr} + \hline + & chisquare.stat & df & p.value \\ + \hline +pt.optim1 & 5.473824 & 6.000000 & 0.484626 \\ + pt.optim2 & 5.329673 & 6.000000 & 0.502277 \\ + pt.optim3 & 5.388158 & 6.000000 & 0.495076 \\ + pt.optim4 & 5.474310 & 6.000000 & 0.484567 \\ + pt.optim5 & 5.337004 & 6.000000 & 0.501372 \\ + pt.optim6 & 5.390662 & 6.000000 & 0.494769 \\ + pt.optim7 & 5.454256 & 6.000000 & 0.487003 \\ + pt.optim8 & 5.476963 & 6.000000 & 0.484245 \\ + \hline +\end{tabular} +\end{table}\end{Schunk} + +Test du $\chi^2$, Méthode avec point de selle +\begin{Schunk} +\begin{Sinput} +> xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),pGAL,"saddlepoint")),digits=6) +\end{Sinput} +% latex table generated in R 3.0.2 by xtable 1.7-1 package +% Sat Mar 15 11:38:58 2014 +\begin{table}[ht] +\centering +\begin{tabular}{rrrr} + \hline + & chisquare.stat & df & p.value \\ + \hline +pt.optim1 & 9.293574 & 6.000000 & 0.157728 \\ + pt.optim2 & 8.345592 & 6.000000 & 0.213862 \\ + pt.optim3 & 9.050625 & 6.000000 & 0.170751 \\ + pt.optim4 & 9.292836 & 6.000000 & 0.157767 \\ + pt.optim5 & 8.344140 & 6.000000 & 0.213959 \\ + pt.optim6 & 9.062381 & 6.000000 & 0.170100 \\ + pt.optim7 & 8.616379 & 6.000000 & 0.196330 \\ + pt.optim8 & 8.610490 & 6.000000 & 0.196698 \\ + \hline +\end{tabular} +\end{table}\end{Schunk} + +Statistique de Kolmogorov-Smirnov +\begin{Schunk} +\begin{Sinput} +> ks.test1 <- function(param,x,y) ks.test(x,y,param) +> xtable(do.call(rbind,mclapply(l.pts.estim,ks.test1,sRET,"pGAL")),digits=6) +\end{Sinput} +% latex table generated in R 3.0.2 by xtable 1.7-1 package +% Sat Mar 15 11:38:58 2014 +\begin{table}[ht] +\centering +\begin{tabular}{rrrrrr} + \hline + & statistic & p.value & alternative & method & data.name \\ + \hline +pt.optim1 & 0.158220 & 0.171912 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim2 & 0.140346 & 0.289345 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim3 & 0.156772 & 0.179751 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim4 & 0.158159 & 0.172235 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim5 & 0.139916 & 0.292753 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim6 & 0.156960 & 0.178718 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim7 & 0.141230 & 0.282437 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + pt.optim8 & 0.140016 & 0.291954 & two-sided & One-sample Kolmogorov-Smirnov test & x \\ + \hline +\end{tabular} +\end{table}\end{Schunk} + +Statistique de distance minimale + +\begin{Schunk} +\begin{Sinput} +> tvariate1 <- seq(-.1,.1,by=0.01) +> xtable(do.call(rbind,mclapply(l.pts.estim,md.test,sRET,tvariate1,cfGAL,empCF)),digits=6) +\end{Sinput} +% latex table generated in R 3.0.2 by xtable 1.7-1 package +% Sat Mar 15 11:38:58 2014 +\begin{table}[ht] +\centering +\begin{tabular}{rrrr} + \hline + & md.stat & df & p.value \\ + \hline +pt.optim1 & 0.000422 & 21.000000 & 0.000000 \\ + pt.optim2 & 0.120174 & 21.000000 & 0.000000 \\ + pt.optim3 & 0.001384 & 21.000000 & 0.000000 \\ + pt.optim4 & 0.000388 & 21.000000 & 0.000000 \\ + pt.optim5 & 0.123295 & 21.000000 & 0.000000 \\ + pt.optim6 & 0.001451 & 21.000000 & 0.000000 \\ + pt.optim7 & 0.007980 & 21.000000 & 0.000000 \\ + pt.optim8 & 0.010416 & 21.000000 & 0.000000 \\ + \hline +\end{tabular} +\end{table}\end{Schunk} + + + + + \end{document} diff --git a/Rplots.pdf b/Rplots.pdf new file mode 100644 index 0000000..92bead9 --- /dev/null +++ b/Rplots.pdf @@ -0,0 +1,1487 @@ +%PDF-1.4 +%ρ\r +1 0 obj +<< +/CreationDate (D:20140315113858) +/ModDate (D:20140315113858) +/Title (R Graphics Output) +/Producer (R 3.0.2) +/Creator (R) +>> +endobj +2 0 obj +<< /Type /Catalog /Pages 3 0 R >> +endobj +7 0 obj +<< /Type /Page /Parent 3 0 R /Contents 8 0 R /Resources 4 0 R >> +endobj +%PDF-1.4 +%ρ\r +1 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