data<-c(27, 82, 115, 126, 155, 161, 243, 294, 340, 384,
457, 680, 855, 877, 974, 1193, 1340, 1884, 2558, 15743)

data



#a+c) maximalizujme logaritmickou verohodnost

logverohodnost<-function(x){
logverohodnost<-sum(dlnorm(data,x[1],x[2],log=TRUE))
}

opt<-optim(c(4,1),logverohodnost,control=list(fnscale=-1),hessian=TRUE)
opt

#MLE
muhat=opt$par[1]
sigmahat=opt$par[2]
muhat
sigmahat

#Odhady podle vzorce
m<-mean(log(data))
s<-sqrt(mean((log(data)-m)^2))
m
s

#b+d)
#odhad Fisherovy informacni matice pro cely vektor (X1,...,Xn), tj. n*J(theta_hat)
I<- -opt$hessian
I

#odhad variancni matice numericky
round(solve(I),4)

#odhad variancni matice ze vzorce
Ihat<-matrix(c(sigmahat^2/20,0,0,sigmahat^2/40),nrow=2)
Ihat

# IS pro mu je

muhat-sqrt(solve(I)[1,1])*qnorm(0.975)
muhat+sqrt(solve(I)[1,1])*qnorm(0.975)

# IS pro sigma je

sigmahat-sqrt(solve(I)[2,2])*qnorm(0.975)
sigmahat+sqrt(solve(I)[2,2])*qnorm(0.975)


###############################
#2.)
#a)
exp(muhat+sigmahat^2/2)

#b)
v<-c(exp(muhat+sigmahat^2/2), exp(muhat+sigmahat^2/2)*sigmahat)%*%solve(I)%*%c(exp(muhat+sigmahat^2/2), exp(muhat+sigmahat^2/2)*sigmahat)
v

exp(muhat+sigmahat^2/2)-sqrt(v)*qnorm(0.975)
exp(muhat+sigmahat^2/2)+sqrt(v)*qnorm(0.975)