cov<-matrix(c(0.09,0.0045,0.0045,0.0225),nrow=2) rematr<-matrix(c(0.2,0.12),ncol=1) ematr<-matrix(c(1,1),ncol=1) nicov<-solve(cov) ttrematr<-t(rematr) ttematr<-t(ematr) A<-ttematr%*%nicov%*%ematr B<-ttematr%*%nicov%*%rematr C<-ttrematr%*%nicov%*%rematr a<-A/(A*C-B*B) b<--2*B/(A*C-B*B) c<-C/(A*C-B*B) ssigma<-function(u) { sigma<-sqrt(a*u*u+b*u+c) return(sigma) } v<-c() u <- seq(0.1,0.2,0.001) for(n in u) { v<-c(v,ssigma(n)) } um<-B/A sigmam<-1/sqrt(A) u_pos <- u[match(min(v),v):length(u)] v_pos <- v[match(min(v),v):length(u)] plot(v_pos,u_pos,type="l",xlim=c(0,0.4),ylim=c(0,0.25),xlab="σ",ylab="μ",col="blue",cex.axis=0.8,lwd=1.5) ustar<--(2*c+b*0.08)/(2*a*0.08+b) sigmastar<-sqrt(a*ustar*ustar+b*ustar+c) lines(c(0,sigmastar),c(0.08,ustar)) points(sigmastar,ustar,col="red",pch=18,cex=1.2) abline(v=seq(0,0.4,0.05),lty=3) abline(h=seq(0,0.25,0.05),lty=3)