714 lines
22 KiB
TeX
714 lines
22 KiB
TeX
\documentclass[letter]{report}
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\usepackage[margin=0.5in]{geometry}
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\usepackage{Sweave}
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\usepackage{graphicx}
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\usepackage[francais]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{verbatim}
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\usepackage{float}
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\usepackage{hyperref}
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\usepackage{scrtime}
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\begin{document}
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\title{GAL Buckle 95}
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\author{François Pelletier}
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\maketitle
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\tableofcontents
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\chapter{Initialisation}
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\section{Chargement des paquets}
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\begin{Schunk}
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\begin{Sinput}
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> setwd("~/maitrise/GAL-Buckle95/")
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> library(actuar)
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> library(MASS)
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> library(xtable)
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> library(parallel)
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> library(moments)
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> library(FourierStuff)
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> library(GeneralizedAsymmetricLaplace)
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> library(GMMStuff)
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> library(OptionPricingStuff)
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> library(QuadraticEstimatingEquations)
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\end{Sinput}
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\end{Schunk}
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\section{Constantes et données}
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\begin{Schunk}
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\begin{Sinput}
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> #Nombre de décimales affichées
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> options(digits=6)
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> #Marge pour intervalles de confiance
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> alpha.confint <- 0.05
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> #Marge pour test d'hypothèses
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> alpha.test <- 0.05
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> #Chargement des données
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> RETURNS <- head(read.csv("abbeyn.csv",sep="\t",header=TRUE)[,1],-1)
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> #Taille de l'échantillon
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> n <- length(RETURNS)
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> #Nom de l'échantillon
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> strData <- "Buckle95"
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\end{Sinput}
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\end{Schunk}
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\section{Test de normalité}
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\begin{Schunk}
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\begin{Sinput}
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> EppsPulley.test(RETURNS)
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\end{Sinput}
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\begin{Soutput}
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Epps-Pulley Normality test
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T: 0.626033
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T*: 0.635568
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p-value: 0.007178
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$Tstat
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[1] 0.626033
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$Tmod
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[1] 0.635568
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$Zscore
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[1] 2.44824
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$Pvalue
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[1] 0.00717788
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$Reject
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[1] TRUE
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\end{Soutput}
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\end{Schunk}
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\chapter{Estimation}
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\section{Données mises à l'échelle}
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\begin{Schunk}
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\begin{Sinput}
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> sRET <- as.vector(scale(RETURNS))
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\end{Sinput}
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\end{Schunk}
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\section{Première estimation par QEE}
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\begin{Schunk}
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\begin{Sinput}
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> ## Point de départ
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> pt.depart <- startparamGAL(sRET)
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> ## Fonctions pour les moments
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> meanQEE <- function(param) mGAL(param,1)
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> varianceQEE <- function(param) cmGAL(param,2)
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> sdQEE <- function(param) sqrt(cmGAL(param,2))
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> skewnessQEE <- function(param) cmGAL(param,3)
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> kurtosisQEE <- function(param) cmGAL(param,4)
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> ## Fonctions pour les dérivées
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> dmeanQEE <- function(param) dmGAL(param,1)
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> dsdQEE <- function(param) dmGAL(param,2)
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> ## Estimation gaussienne
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> optim1 <- optim(pt.depart,obj.gauss,gr=NULL,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE)
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> pt.optim1 <- optim1$par
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> ## Estimation de crowder
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> optim2 <- optim(pt.depart,obj.Crowder,gr=NULL,sRET,
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+ meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE)
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> pt.optim2 <- optim2$par
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> ## Estimation de crowder modifiée
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> optim3 <- optim(pt.depart,obj.Crowder.Mod,gr=NULL,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE)
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> pt.optim3 <- optim3$par
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\end{Sinput}
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\end{Schunk}
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\section{Résultats de la première estimation par QEE}
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\begin{Schunk}
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\begin{Sinput}
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> cov.optim1 <- covariance.QEE(M.gauss(pt.optim1,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE),
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+ V.gauss(pt.optim1,sRET,meanQEE,varianceQEE,
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+ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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> cov.optim2 <- covariance.QEE(M.Crowder(pt.optim2,sRET,
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+ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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+ V.Crowder(pt.optim2,sRET,varianceQEE,
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+ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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> cov.optim3 <- covariance.QEE(M.Crowder.Mod(pt.optim3,sRET,
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+ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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+ V.Crowder.Mod(pt.optim3,sRET,varianceQEE,dmeanQEE,dsdQEE),n)
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> confidence.interval.QEE(pt.optim1,cov.optim1,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.780018 -0.726048 -0.672077
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[2,] 0.436002 0.596316 0.756630
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[3,] 0.262650 0.359186 0.455722
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[4,] 1.994757 2.021370 2.047982
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\end{Soutput}
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\begin{Sinput}
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> confidence.interval.QEE(pt.optim2,cov.optim2,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.694457 -0.627404 -0.560351
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[2,] 0.413764 0.640292 0.866820
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[3,] 0.232650 0.334028 0.435405
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[4,] 1.839966 1.878296 1.916626
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\end{Soutput}
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\begin{Sinput}
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> confidence.interval.QEE(pt.optim3,cov.optim3,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.765288 -0.711439 -0.657589
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[2,] 0.455485 0.606642 0.757798
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[3,] 0.264669 0.362932 0.461195
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[4,] 1.932691 1.960299 1.987906
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\end{Soutput}
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\end{Schunk}
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\section{Seconde estimation par QEE}
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\begin{Schunk}
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\begin{Sinput}
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> ## Estimation gaussienne
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> optim4 <- optim(pt.optim1,obj.gauss,gr=NULL,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE,
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+ ginv(V.gauss(pt.optim1,sRET,meanQEE,
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+ varianceQEE,skewnessQEE,kurtosisQEE,
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+ dmeanQEE,dsdQEE)))
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> pt.optim4 <- optim4$par
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> ## Estimation de crowder
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> optim5 <- optim(pt.optim2,obj.Crowder,gr=NULL,sRET,
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+ meanQEE,varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE,
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+ ginv(V.Crowder(pt.optim2,sRET,varianceQEE,skewnessQEE,
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+ kurtosisQEE,dmeanQEE,dsdQEE)))
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> pt.optim5 <- optim5$par
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> ## Estimation de crowder modifiée
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> optim6 <- optim(pt.optim3,obj.Crowder.Mod,gr=NULL,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE,
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+ ginv(V.Crowder.Mod(pt.optim3,sRET,varianceQEE,
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+ dmeanQEE,dsdQEE)))
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> pt.optim6 <- optim6$par
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\end{Sinput}
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\end{Schunk}
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\section{Résultats de la seconde estimation par QEE}
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\begin{Schunk}
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\begin{Sinput}
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> cov.optim4 <- covariance.QEE(M.gauss(pt.optim4,sRET,
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+ meanQEE,varianceQEE,dmeanQEE,dsdQEE),
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+ V.gauss(pt.optim4,sRET,meanQEE,varianceQEE,
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+ skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),n)
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> cov.optim5 <- covariance.QEE(M.Crowder(pt.optim5,sRET,
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+ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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+ V.Crowder(pt.optim5,sRET,varianceQEE,skewnessQEE,
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+ kurtosisQEE,dmeanQEE,dsdQEE),n)
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> cov.optim6 <- covariance.QEE(M.Crowder.Mod(pt.optim6,sRET,
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+ varianceQEE,skewnessQEE,kurtosisQEE,dmeanQEE,dsdQEE),
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+ V.Crowder.Mod(pt.optim6,sRET,varianceQEE,dmeanQEE,dsdQEE),n)
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> confidence.interval.QEE(pt.optim4,cov.optim4,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.779792 -0.725853 -0.671914
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[2,] 0.436017 0.596319 0.756622
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[3,] 0.262456 0.358969 0.455482
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[4,] 1.995452 2.022048 2.048644
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\end{Soutput}
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\begin{Sinput}
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> confidence.interval.QEE(pt.optim5,cov.optim5,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.692712 -0.625874 -0.559036
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[2,] 0.414139 0.640445 0.866750
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[3,] 0.231568 0.332845 0.434122
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[4,] 1.842116 1.880376 1.918636
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\end{Soutput}
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\begin{Sinput}
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> confidence.interval.QEE(pt.optim6,cov.optim6,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.766288 -0.712450 -0.658612
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[2,] 0.455051 0.606193 0.757334
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[3,] 0.264972 0.363196 0.461419
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[4,] 1.934050 1.961614 1.989178
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\end{Soutput}
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\end{Schunk}
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\section{Estimation par GMM}
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\begin{Schunk}
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\begin{Sinput}
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> ## GMM régulier
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> optim7 <- optim.GMM(pt.depart,
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+ conditions.vector=meanvariance.gmm.vector,
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+ data=sRET,W=diag(2),
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+ meanf=meanQEE,variancef=varianceQEE)
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> pt.optim7 <- optim7$par
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> cov.optim7 <- mean.variance.GMM.gradient.GAL(pt.optim7,sRET) %*%
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+ covariance.GMM(pt.optim7,meanvariance.gmm.vector,sRET,
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+ meanf=meanQEE,variancef=varianceQEE) %*%
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+ t(mean.variance.GMM.gradient.GAL(pt.optim7,sRET)) / n
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> ## GMM itératif
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> optim8 <- iterative.GMM(pt.depart,
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+ conditions.vector=meanvariance.gmm.vector,
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+ data=sRET,W=diag(2),
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+ meanf=meanQEE,variancef=varianceQEE)
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> pt.optim8 <- optim8$par
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> cov.optim8 <- mean.variance.GMM.gradient.GAL(pt.optim8,sRET) %*%
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+ optim8$cov %*%
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+ t(mean.variance.GMM.gradient.GAL(pt.optim8,sRET)) / n
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> confidence.interval.QEE(pt.optim7,cov.optim7,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.878702 -0.641646 -0.404589
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[2,] -0.469225 0.625908 1.721040
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[3,] -0.192234 0.326366 0.844965
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[4,] 1.696121 1.965995 2.235869
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\end{Soutput}
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\begin{Sinput}
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> confidence.interval.QEE(pt.optim8,cov.optim8,n)
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\end{Sinput}
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\begin{Soutput}
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LOWER ESTIMATE UPPER
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[1,] -0.874031 -0.636980 -0.399929
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[2,] -0.473292 0.626346 1.725984
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[3,] -0.193600 0.322895 0.839390
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[4,] 1.704166 1.972716 2.241265
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\end{Soutput}
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\end{Schunk}
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\chapter{Comparaison des résultats}
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\begin{Schunk}
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\begin{Sinput}
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> # Aggrégation des estimateurs (pour simplifier les calculs)
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> pts.estim <- cbind(pt.optim1,pt.optim2,pt.optim3,pt.optim4,
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+ pt.optim5,pt.optim6,pt.optim7,pt.optim8)
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> l.pts.estim <- as.list(data.frame(pts.estim))
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\end{Sinput}
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\end{Schunk}
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\section{Fonction de répartition}
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\begin{Schunk}
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\begin{Sinput}
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> # Points d'évaluation
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> xi <- seq(2*min(sRET),2*max(sRET),length.out=2^6)
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> # Fonction de répartition par intégration de la fonction caractéristique
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> dist1 <- cbind(cftocdf(xi,cfGAL,param=pt.optim1),
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+ cftocdf(xi,cfGAL,param=pt.optim2),
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+ cftocdf(xi,cfGAL,param=pt.optim3),
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+ cftocdf(xi,cfGAL,param=pt.optim4),
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+ cftocdf(xi,cfGAL,param=pt.optim5),
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+ cftocdf(xi,cfGAL,param=pt.optim6),
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+ cftocdf(xi,cfGAL,param=pt.optim7),
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+ cftocdf(xi,cfGAL,param=pt.optim8))
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> # Fonction de répartition par point de selle
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> dist2 <- cbind(psaddleapproxGAL(xi,pt.optim1),
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+ psaddleapproxGAL(xi,pt.optim2),
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+ psaddleapproxGAL(xi,pt.optim3),
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+ psaddleapproxGAL(xi,pt.optim4),
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+ psaddleapproxGAL(xi,pt.optim5),
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+ psaddleapproxGAL(xi,pt.optim6),
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+ psaddleapproxGAL(xi,pt.optim7),
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+ psaddleapproxGAL(xi,pt.optim8))
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> # Fonction de répartition par intégration de la fonction de densité
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> dist3 <- cbind(pGAL(xi,pt.optim1),
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+ pGAL(xi,pt.optim2),
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+ pGAL(xi,pt.optim3),
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+ pGAL(xi,pt.optim4),
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+ pGAL(xi,pt.optim5),
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+ pGAL(xi,pt.optim6),
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+ pGAL(xi,pt.optim7),
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+ pGAL(xi,pt.optim8))
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\end{Sinput}
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\end{Schunk}
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\pagebreak
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\subsection{Graphiques}
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\begin{Schunk}
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\begin{Sinput}
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> for (i in 1:8)
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+ {
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+ file<-paste(strData,"-repart-",i,".pdf",sep="")
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+ pdf(file=file,paper="special",width=6,height=6)
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+ plot.ecdf(sRET,main=paste("Fonction de répartition ",i))
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+ lines(xi,dist1[,i],col="green")
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+ lines(xi,dist2[,1],col="red")
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+ lines(xi,dist3[,1],col="pink")
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+ lines(xi,pnorm(xi),type="l",col="blue")
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+ dev.off()
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+ cat("\\includegraphics[height=4in,width=4in]{",
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+ file,"}\n",sep="")
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+ }
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\end{Sinput}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-1.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-2.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-3.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-4.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-5.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-6.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-7.pdf}
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\includegraphics[height=4in,width=4in]{Buckle95-repart-8.pdf}\end{Schunk}
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\subsection{Statistiques}
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Test du $\chi^2$, Méthode avec intégration
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\begin{Schunk}
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\begin{Sinput}
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> chisquare.test1 <- function(param,DATA.hist,FUN,method)
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+ {
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+ chisquare.test(DATA.hist,FUN,param,method=method)
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+ }
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> xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),
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+ cfGAL,"integral")),digits=6)
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\end{Sinput}
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% latex table generated in R 3.1.0 by xtable 1.7-3 package
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% Sat May 24 10:56:20 2014
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\begin{table}[ht]
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\centering
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\begin{tabular}{rrrr}
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\hline
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& chisquare.stat & df & p.value \\
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\hline
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pt.optim1 & 5.473824 & 6.000000 & 0.484626 \\
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pt.optim2 & 5.329673 & 6.000000 & 0.502277 \\
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pt.optim3 & 5.388158 & 6.000000 & 0.495076 \\
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pt.optim4 & 5.474310 & 6.000000 & 0.484567 \\
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pt.optim5 & 5.337004 & 6.000000 & 0.501372 \\
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pt.optim6 & 5.390662 & 6.000000 & 0.494769 \\
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pt.optim7 & 5.454256 & 6.000000 & 0.487003 \\
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pt.optim8 & 5.476963 & 6.000000 & 0.484245 \\
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\hline
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\end{tabular}
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\end{table}\end{Schunk}
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Test du $\chi^2$, Méthode avec point de selle
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\begin{Schunk}
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\begin{Sinput}
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> xtable(do.call(rbind,lapply(l.pts.estim,chisquare.test1,hist(sRET),
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+ pGAL,"saddlepoint")),digits=6)
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\end{Sinput}
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% latex table generated in R 3.1.0 by xtable 1.7-3 package
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% Sat May 24 10:56:20 2014
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\begin{table}[ht]
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\centering
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\begin{tabular}{rrrr}
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\hline
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& chisquare.stat & df & p.value \\
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\hline
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pt.optim1 & 9.293574 & 6.000000 & 0.157728 \\
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pt.optim2 & 8.345592 & 6.000000 & 0.213862 \\
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pt.optim3 & 9.050625 & 6.000000 & 0.170751 \\
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|
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 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 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 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 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}
|
|
|
|
\section{Fonction de densité}
|
|
|
|
Intégration de la fonction de densité approximée avec le point de selle, pour la
|
|
normaliser en fonction qui intègre à 1.
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> f_integrale_saddle <- function(param,f,lower,upper)
|
|
+ integrate(f,lower,upper,param)$value
|
|
> norm_int_saddle <- sapply(l.pts.estim,f_integrale_saddle,
|
|
+ f=dsaddleapproxGAL,lower=-Inf,upper=Inf)
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
Séquence de points pour les graphiques
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> x_sRET <- seq(min(sRET)-sd(sRET),max(sRET)+sd(sRET),length.out=50)
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
Graphique de la fonction de densité
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> colors2=c("black","red","green","blue","grey")
|
|
> for (i in 1:dim(pts.estim)[2])
|
|
+ {
|
|
+ file=paste(strData,"-densite-", i, ".pdf", sep="")
|
|
+ pdf(file=file, paper="special", width=6, height=6)
|
|
+ plot(density(sRET),ylim=c(0,.7),type="l",
|
|
+ main=paste("Densité de",strData, i),xlab=strData,
|
|
+ ylab="f",lwd=2,lty=1)
|
|
+ points(x_sRET,
|
|
+ dGAL(x_sRET,pts.estim[,i]),
|
|
+ type="b",ylim=c(0,4),col="red",pch=19,lwd=2,lty=2)
|
|
+ points(x_sRET,
|
|
+ dsaddleapproxGAL(x_sRET,pts.estim[,i])/norm_int_saddle[i],
|
|
+ type="b",ylim=c(0,4),col="green",pch=20,lwd=2,lty=3)
|
|
+
|
|
+ lines(x_sRET,dnorm(x_sRET),type="b",col="blue",
|
|
+ pch=21,lwd=2,lty=4)
|
|
+ points(seq(-2,4,length.out=1000)[seq(40,1000,by=40)],
|
|
+ cftodensity.fft(cfGAL,1000,-2,4,pts.estim[,i])$dens[seq(40,1000,by=40)],
|
|
+ type="b",col="grey",pch=23,lty=6)
|
|
+ legend(quantile(sRET,0.9),0.7, c("emp","est.GAL","pt.selle","appx.nrm","fft"),
|
|
+ cex=0.8, col=colors2, pch=c(NA,19:23), lty=1:6, title="Courbes")
|
|
+ dev.off()
|
|
+ cat("\\includegraphics[height=4in, width=4in]{"
|
|
+ ,file, "}\n", sep="")
|
|
+ }
|
|
\end{Sinput}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-1.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-2.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-3.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-4.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-5.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-6.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-7.pdf}
|
|
\includegraphics[height=4in, width=4in]{Buckle95-densite-8.pdf}\end{Schunk}
|
|
|
|
\subsection{Tests avec contraintes}
|
|
|
|
Test de Wald
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> R <- matrix(c(0,0,1,0,
|
|
+ 0,0,0,1),ncol=4)
|
|
> r <- matrix(c(0,0),ncol=1)
|
|
> V <- lapply(l.pts.estim,covariance.GMM,meanvariance.gmm.vector,
|
|
+ sRET,meanQEE,varianceQEE)
|
|
> D <- lapply(l.pts.estim,mean.variance.GMM.gradient.GAL,sRET)
|
|
> xtable(mapply(Wald.Test,l.pts.estim,n,list(R),list(r),V,D),
|
|
+ caption="Test de Wald", digits=2)
|
|
\end{Sinput}
|
|
\begin{Soutput}
|
|
% latex table generated in R 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 2014
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{rrrrrrrrr}
|
|
\hline
|
|
& pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\
|
|
\hline
|
|
wald.stat & 1861.21 & 1796.75 & 1690.26 & 1865.01 & 1814.81 & 1690.62 & 2111.08 & 2175.45 \\
|
|
p.value & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
|
|
reject & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 & 1.00 \\
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Test de Wald}
|
|
\end{table}
|
|
\end{Soutput}
|
|
\end{Schunk}
|
|
|
|
\subsection{Vrais paramètres}
|
|
|
|
Comme nous avons estimé avec des données centrées et réduites, nous utilisons
|
|
une propriété de la distribution GAL qui nous permet d'obtenir les paramètres
|
|
des rendements non réduits.
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> pts.estim.ns <- apply(pts.estim,2,scaleGAL,type="mu",
|
|
+ mean(RETURNS),sd(RETURNS))
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> xtable(pts.estim.ns,
|
|
+ caption="Paramètres des données non centrées et réduites",
|
|
+ digits=4)
|
|
\end{Sinput}
|
|
% latex table generated in R 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 2014
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{rrrrrrrrr}
|
|
\hline
|
|
& pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\
|
|
\hline
|
|
1 & -0.0092 & -0.0080 & -0.0090 & -0.0092 & -0.0079 & -0.0091 & -0.0081 & -0.0081 \\
|
|
2 & 0.0078 & 0.0083 & 0.0079 & 0.0078 & 0.0083 & 0.0079 & 0.0081 & 0.0081 \\
|
|
3 & 0.0033 & 0.0031 & 0.0033 & 0.0033 & 0.0031 & 0.0033 & 0.0030 & 0.0030 \\
|
|
4 & 2.0214 & 1.8783 & 1.9603 & 2.0220 & 1.8804 & 1.9616 & 1.9660 & 1.9727 \\
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Paramètres des données non centrées et réduites}
|
|
\end{table}\end{Schunk}
|
|
|
|
\section{Prix d'options}
|
|
|
|
\subsection{Données de base}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> #Taux sans risque
|
|
> rfrate <- .05/365
|
|
> #Échéance
|
|
> T <- 30
|
|
> #Pas de discrétisation courbe des prix
|
|
> pas <- 0.005
|
|
> #Prix initial
|
|
> stock0 <- 299
|
|
> #Prix d'exercice dans le cours (put)
|
|
> strike1 <- stock0*seq(0.98,1,pas)
|
|
> #Prix d'exercice hors le cours (put)
|
|
> strike2 <- stock0*seq(1+pas,1.02,pas)
|
|
> #Prix d'exercice combinés
|
|
> strike <- c(strike1,strike2)
|
|
> #Damping parameter
|
|
> alpha <- 3
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
\subsection{Paramètres neutres au risque}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> pts.estim.ns.rn <- apply(pts.estim.ns,2,riskneutralparGAL,rfrate)
|
|
> l.pts.estim.ns.rn <- as.list(data.frame(pts.estim.ns.rn))
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> xtable(pts.estim.ns.rn,caption="Paramètres neutres au risque",digits=4)
|
|
\end{Sinput}
|
|
% latex table generated in R 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 2014
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{rrrrrrrrr}
|
|
\hline
|
|
& pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\
|
|
\hline
|
|
1 & -0.0066 & -0.0057 & -0.0065 & -0.0066 & -0.0057 & -0.0065 & -0.0058 & -0.0058 \\
|
|
2 & 0.0078 & 0.0083 & 0.0079 & 0.0078 & 0.0083 & 0.0079 & 0.0081 & 0.0081 \\
|
|
3 & 0.0033 & 0.0031 & 0.0033 & 0.0033 & 0.0031 & 0.0033 & 0.0030 & 0.0030 \\
|
|
4 & 2.0214 & 1.8783 & 1.9603 & 2.0220 & 1.8804 & 1.9616 & 1.9660 & 1.9727 \\
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Paramètres neutres au risque}
|
|
\end{table}\end{Schunk}
|
|
|
|
|
|
\subsection{Méthode de Epps}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> f_putEpps <- function(param,strikeprice,char.fn,eval.time,expiry.time,rate,...)
|
|
+ putEpps(strikeprice,char.fn,param,eval.time,expiry.time,rate,...)
|
|
> prix_Epps <- as.data.frame(sapply(l.pts.estim.ns.rn,f_putEpps,strike/stock0,cfLM,0,T,rfrate))
|
|
\end{Sinput}
|
|
\end{Schunk}
|
|
|
|
\begin{Schunk}
|
|
\begin{Sinput}
|
|
> xtable(prix_Epps,caption="Prix unitaire de l'option de vente, Méthode de Epps",digits=6)
|
|
\end{Sinput}
|
|
% latex table generated in R 3.1.0 by xtable 1.7-3 package
|
|
% Sat May 24 10:56:20 2014
|
|
\begin{table}[ht]
|
|
\centering
|
|
\begin{tabular}{rrrrrrrrr}
|
|
\hline
|
|
& pt.optim1 & pt.optim2 & pt.optim3 & pt.optim4 & pt.optim5 & pt.optim6 & pt.optim7 & pt.optim8 \\
|
|
\hline
|
|
1 & 0.015415 & 0.015785 & 0.015431 & 0.015417 & 0.015794 & 0.015427 & 0.015792 & 0.015819 \\
|
|
2 & 0.017360 & 0.017737 & 0.017377 & 0.017362 & 0.017746 & 0.017372 & 0.017742 & 0.017770 \\
|
|
3 & 0.019456 & 0.019838 & 0.019474 & 0.019458 & 0.019847 & 0.019469 & 0.019843 & 0.019871 \\
|
|
4 & 0.021705 & 0.022090 & 0.021724 & 0.021707 & 0.022099 & 0.021719 & 0.022093 & 0.022122 \\
|
|
5 & 0.024107 & 0.024492 & 0.024126 & 0.024109 & 0.024501 & 0.024121 & 0.024495 & 0.024523 \\
|
|
6 & 0.026661 & 0.027044 & 0.026681 & 0.026663 & 0.027053 & 0.026676 & 0.027046 & 0.027074 \\
|
|
7 & 0.029365 & 0.029745 & 0.029385 & 0.029367 & 0.029753 & 0.029381 & 0.029745 & 0.029773 \\
|
|
8 & 0.032217 & 0.032591 & 0.032238 & 0.032219 & 0.032600 & 0.032233 & 0.032591 & 0.032617 \\
|
|
9 & 0.035214 & 0.035580 & 0.035235 & 0.035216 & 0.035589 & 0.035231 & 0.035579 & 0.035605 \\
|
|
\hline
|
|
\end{tabular}
|
|
\caption{Prix unitaire de l'option de vente, Méthode de Epps}
|
|
\end{table}\end{Schunk}
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
\end{document}
|