#  MLE for double expon.

X <- c(-1.3, -0.9, -0.5, -0.1, 0.2, 0.8, 1.2)   # some "data"
n <- length(X)

loglik <- function(theta){
  - sum(abs(X-theta))
}
  
xc <- seq(-1,1, by = 0.01)
yc <- 0*xc

for (i in 1:length(xc)){
   yc[i] <- loglik(xc[i])
}

plot(xc, yc)


n <- 5000
X <- rexp(n,1) *  sample(c(-1,1), n, replace= TRUE)   
   # assume theta_0 = 0

showX <- pmax(pmin(X,5),-5)
hist(showX, breaks = seq(-5,5,by = 0.2))


xc <- seq(-0.1,0.1, by = 0.001)
yc <- 0*xc

for (i in 1:length(xc)){
   yc[i] <- loglik(xc[i])
}

plot(xc, yc,xlim=c(-0.05,0.05))
# plot(xc, yc)

  that <- median(X)
  yc2 <- loglik(that) - n/2*(xc - that)^2

  lines(xc, yc2, col="red" , lwd=2)  # overlays plot with the Fisher infromation-based parabola






## MLE for Cauchy: problem 6.1.8 

 X <- c(-1.94, 0.59, -5.98, -0.08, -0.77)

 n <- length(X)
  
 loglikn <- function(th){     # negative log-lkhd, to use in Nelder-Mead
   sum( log (1+(X-th)^2) )
}

 xc <- seq (-2,2,0.1) 
 yc <- 0*xc

 for (i in 1:length(xc)){
   yc[i] <- -loglikn(xc[i]) 
 }

 plot(xc,yc, type="l") 

 optim(0,loglikn)    # this gives a numerical value for MLE


# -------------------------------------------------------------------

 n <- 50
 X <- rt(n,df = 1)    # random sample from Cauchy
 

xc <- seq (-1,1,0.01) 
 yc <- 0*xc

 for (i in 1:length(xc)){
   yc[i] <- -loglikn(xc[i]) 
 }

 plot(xc,yc, type="l", lwd=2)   # this represents the log-likhd curve when n is "large"