lecauchy=function(x,toler=.001){      #x is a vector here
  startvalue=median(x)
  n=length(x);
  thetahatcurr=startvalue;
  # Compute first deriviative of log likelihood
  firstderivll=2*sum((x-thetahatcurr)/(1+(x-thetahatcurr)^2))
  # Continue Newton’s method until the first derivative
  # of the likelihood is within toler of 0.001
  while(abs(firstderivll)>toler){
    # Compute second derivative of log likelihood
    secondderivll=2*sum(((x-thetahatcurr)^2-1)/(1+(x-thetahatcurr)^2)^2);
    # Newton’s method update of estimate of theta
    thetahatnew=thetahatcurr-firstderivll/secondderivll;
    thetahatcurr=thetahatnew;
    # Compute first derivative of log likelihood
    firstderivll=2*sum((x-thetahatcurr)/(1+(x-thetahatcurr)^2))
  }
  list(thetahat=thetahatcurr);
}


plotlikecauchy=function(x){
  tht=seq(0,200, by = 0.1)
  li=numeric(length(tht))
  for (i in 1:length(tht)){
  li[i] = prod(1/(1+(x-tht[i])^2));
  }
  plot(tht,log(li), type="l")
  return(tht[which.max(log(li))])
  
  
}

data=c(221,171,198,189,189,135, 162, 135,117,162)