CDF and PDF definitions
CDFxMarketLN = function(sigma,gama,mu,s, x){
pnorm((log(x-gama)-mu)/(sigma*s),0,1)
}
CDFxMarketTLN = function(sigma,gama,mu,s,xbar,x){
(pnorm((log(x-gama)-mu)/(sigma*s),0,1)-pnorm((log(xbar-gama)-mu)/(sigma*s),0,1))/(1-pnorm((log(xbar-gama)-mu)/(sigma*s),0,1))
}
PDFxMarketTLN=function(sigma,gama,mu,s,xbar,x){
(1/(1-pnorm((log(xbar-gama)-mu)/(sigma*s),0,1)))*(exp(-(mu-log(x-gama))^2/(2*s^2*sigma^2))/(sqrt(2*pi)*s*(x-gama)*sigma))
}
CDFxPlanLN = function(sigma,gama,mu,s,omega,x){
pnorm((log((omega^sigma/(1+omega^sigma))*x*(x-gama)^((1-sigma)/sigma))-mu)/(s),0,1)
}
CDFxPlanTLN = function(sigma,gama,mu,s,omega,xbar,x){
(pnorm((log((omega^sigma/(1+omega^sigma))*x*(x-gama)^((1-sigma)/sigma))-mu)/(s),0,1)-pnorm((log((omega^sigma/(1+omega^sigma))*xbar*(xbar-gama)^((1-sigma)/sigma))-mu)/(s),0,1))/(1-pnorm((log((omega^sigma/(1+omega^sigma))*xbar*(xbar-gama)^((1-sigma)/sigma))-mu)/(s),0,1))
}
PDFxPlanTLN = function(sigma,gama,mu,s,omega,xbar, x){
(1/(1-pnorm(((log((omega^sigma/(1+omega^sigma))*xbar*(xbar-gama)^((1-sigma)/sigma))-mu)/s),0,1)))*(exp(-(mu-log((omega^sigma/(1+omega^sigma))*x*(x-gama)^(1/sigma-1)))^2/(2*s^2))*(x-gama*sigma)/(sqrt(2*pi)*s*x*(x-gama)*sigma))
}
PDFdifferenceLN=function(sigma,gama,muM,muP,s,omega,xbar,x){
PDFxPlanTLN(sigma,gama,muP,s,omega,xbar, x)-PDFxMarketTLN(sigma,gama,muM,s,xbar, x)
}
Calculating the intersections between PDFs and shares:
gama1=1
xbar1=gama1*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f1LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama1,muCREMR_lnt+log(gama1),(muCREMR_lnt+log(gama1))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar1,x)
}
uniroot(f1LN,c(1.4,1.5))
## $root
## [1] 1.472774
##
## $f.root
## [1] 1.661879e-05
##
## $iter
## [1] 4
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc1LN<-uniroot(f1LN,c(1.4,1.5),tol = 1e-6)
CDFxMarketTLN(sigmaCREMR_lnt,gama1,muCREMR_lnt+log(gama1),sCREMR_lnt,xbar1,xc1LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama1,(muCREMR_lnt+log(gama1))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar1,xc1LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama1,muCREMR_lnt+log(gama1),sCREMR_lnt,xbar1,xc1LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama1,(muCREMR_lnt+log(gama1))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar1,xc1LN$root)
## [1] 5.22023
gama3=1.5
xbar3=gama3*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f3LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama3,muCREMR_lnt+log(gama3),(muCREMR_lnt+log(gama3))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar3,x)
}
uniroot(f3LN,c(2.2,2.21))
## $root
## [1] 2.209169
##
## $f.root
## [1] 1.706608e-05
##
## $iter
## [1] 2
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc3LN<-uniroot(f3LN,c(2.2,2.21),tol = 1e-6)
CDFxMarketTLN(sigmaCREMR_lnt,gama3,muCREMR_lnt+log(gama3),sCREMR_lnt,xbar3,xc3LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama3,(muCREMR_lnt+log(gama3))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar3,xc3LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama3,muCREMR_lnt+log(gama3),sCREMR_lnt,xbar3,xc3LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama3,(muCREMR_lnt+log(gama3))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar3,xc3LN$root)
## [1] 5.22023
gama5=2
xbar5=gama5*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f5LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama5,muCREMR_lnt+log(gama5),(muCREMR_lnt+log(gama5))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar5,x)
}
uniroot(f5LN,c(2.93,2.95))
## $root
## [1] 2.945531
##
## $f.root
## [1] -2.307234e-08
##
## $iter
## [1] 3
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc5LN<-uniroot(f5LN,c(2.93,2.95))
CDFxMarketTLN(sigmaCREMR_lnt,gama5,muCREMR_lnt+log(gama5),sCREMR_lnt,xbar5,xc5LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama5,(muCREMR_lnt+log(gama5))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar5,xc5LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama5,muCREMR_lnt+log(gama5),sCREMR_lnt,xbar5,xc5LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama5,(muCREMR_lnt+log(gama5))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar5,xc5LN$root)
## [1] 5.22023
gama6=3
xbar6=gama6*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f6LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama6,muCREMR_lnt+log(gama6),(muCREMR_lnt+log(gama6))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar6,x)
}
uniroot(f6LN,c(4.40,4.42))
## $root
## [1] 4.418277
##
## $f.root
## [1] -3.885796e-06
##
## $iter
## [1] 2
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc6LN<-uniroot(f6LN,c(4.40,4.42))
CDFxMarketTLN(sigmaCREMR_lnt,gama6,muCREMR_lnt+log(gama6),sCREMR_lnt,xbar6,xc6LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama6,(muCREMR_lnt+log(gama6))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar6,xc6LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama6,muCREMR_lnt+log(gama6),sCREMR_lnt,xbar6,xc6LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama6,(muCREMR_lnt+log(gama6))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar6,xc6LN$root)
## [1] 5.22023
gama7=4
xbar7=gama7*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f7LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama7,muCREMR_lnt+log(gama7),(muCREMR_lnt+log(gama7))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar7,x)
}
uniroot(f7LN,c(5.8,5.9))
## $root
## [1] 5.89106
##
## $f.root
## [1] -9.493441e-08
##
## $iter
## [1] 3
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc7LN<-uniroot(f7LN,c(5.8,5.9))
CDFxMarketTLN(sigmaCREMR_lnt,gama7,muCREMR_lnt+log(gama7),sCREMR_lnt,xbar7,xc7LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama7,(muCREMR_lnt+log(gama7))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar7,xc7LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama7,muCREMR_lnt+log(gama7),sCREMR_lnt,xbar7,xc7LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama7,(muCREMR_lnt+log(gama7))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar7,xc7LN$root)
## [1] 5.22023
gama8=5
xbar8=gama8*(omegaCREMR_lnt^sigmaCREMR_lnt+1)
f8LN=function(x){
PDFdifferenceLN(sigmaCREMR_lnt,gama8,muCREMR_lnt+log(gama8),(muCREMR_lnt+log(gama8))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar8,x)
}
uniroot(f8LN,c(7.3,7.4))
## $root
## [1] 7.363823
##
## $f.root
## [1] -2.613998e-07
##
## $iter
## [1] 3
##
## $init.it
## [1] NA
##
## $estim.prec
## [1] 6.103516e-05
xc8LN<-uniroot(f8LN,c(7.3,7.4))
CDFxMarketTLN(sigmaCREMR_lnt,gama8,muCREMR_lnt+log(gama8),sCREMR_lnt,xbar8,xc8LN$root)
## [1] 0.7966002
CDFxPlanTLN(sigmaCREMR_lnt,gama8,(muCREMR_lnt+log(gama8))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar8,xc8LN$root)
## [1] 0.1525987
CDFxMarketTLN(sigmaCREMR_lnt,gama8,muCREMR_lnt+log(gama8),sCREMR_lnt,xbar8,xc8LN$root)/CDFxPlanTLN(sigmaCREMR_lnt,gama8,(muCREMR_lnt+log(gama8))/sigmaCREMR_lnt,sCREMR_lnt,omegaCREMR_lnt,xbar8,xc8LN$root)
## [1] 5.22023