### Note: command below would have worked if computer had been connected
### to the internet
> foo <- read.table("http://www.stat.umn.edu/geyer/somedata.txt", header = TRUE)
unable to resolve 'www.stat.umn.edu'.
Error in file(file, "r") : cannot open URL `http://www.stat.umn.edu/geyer/somedata.txt'

> help(nlm)
> mlogl <- function(theta, x) sum(log(1 + (x - theta)^2))
> mlogl2 <- function(theta, x) sum(- dcauchy(x, theta, log = TRUE))
> n <- 30
> x <- rcauchy(n)
> out <- nlm(mlogl, median(x))
Error in log(1 + (x - theta)^2) : Argument "x" is missing, with no default
> out <- nlm(mlogl, median(x), x = x)
> out
$minimum
[1] 52.88128

$estimate
[1] -0.2703223

$gradient
[1] 8.171241e-07

$code
[1] 1

$iterations
[1] 3

> median(x)
[1] -0.2562955
> out <- nlm(mlogl, median(x), x = x, print.level = 2, hessian = TRUE)
iteration = 0
Step:
[1] 0
Parameter:
[1] -0.2562955
Function Value
[1] 52.88282
Gradient:
[1] 0.2193642

iteration = 1
Step:
[1] -0.02193642
Parameter:
[1] -0.2782320
Function Value
[1] 52.88177
Gradient:
[1] -0.1230431

iteration = 2
Step:
[1] 0.007882791
Parameter:
[1] -0.2703492
Function Value
[1] 52.88128
Gradient:
[1] -0.0004175575

iteration = 3
Parameter:
[1] -0.2703223
Function Value
[1] 52.88128
Gradient:
[1] 8.171241e-07

Relative gradient close to zero.
Current iterate is probably solution.

> out
$minimum
[1] 52.88128

$estimate
[1] -0.2703223

$gradient
[1] 8.171241e-07

$hessian
         [,1]
[1,] 15.58790

$code
[1] 1

$iterations
[1] 3

> 1 / sqrt(out$hessian)
          [,1]
[1,] 0.2532831
> 1 / sqrt(out$hessian) #### asymptotic standard dev. of MLE
          [,1]
[1,] 0.2532831
> 
> out2 <- nlm(mlogl2, median(x), x = x)
> out$minimum
[1] 52.88128
> out2$minimum
[1] 87.22318
> out2$estimate
[1] -0.2703223
> out$estimate
[1] -0.2703223

### this where we edited the file to create the function mlogl3

 mlogl3 <- function(theta, x) {
      if(length(theta) != 2) stop("Oops!")
      sum(- dcauchy(x, location = theta[1], scale = theta[2], log = TRUE))
 }
 n <- 30
 x <- rcauchy(n)
 out <- nlm(mlogl, median(x))

### here we go back to printout

>  mlogl3 <- function(theta, x) {
+       if(length(theta) != 2) stop("Oops!")
+       sum(- dcauchy(x, location = theta[1], scale = theta[2], log = TRUE))
+  }
>  n <- 30
>  x <- rcauchy(n)
> mlogl3
function(theta, x) {
      if(length(theta) != 2) stop("Oops!")
      sum(- dcauchy(x, location = theta[1], scale = theta[2], log = TRUE))
 }
> qcauchy(3/4) - qcauchy(1/4)
[1] 2
> theta.start <- c(median(x), IQR(x) / 2)
> theta.start
[1] 0.004699155 1.029976175
>  x <- rcauchy(n)
> theta.start <- c(median(x), IQR(x) / 2)
> theta.start
[1] -0.5334394  1.0217893
> nlm(mlogl3, theta.start, hessian = TRUE)
Error in dcauchy(x, location = theta[1], scale = theta[2], log = TRUE) : 
        Argument "x" is missing, with no default
> nlm(mlogl3, theta.start, hessian = TRUE, x = x)
$minimum
[1] 77.25726

$estimate
[1] -0.3856592  1.2444707

$gradient
[1] -2.33058e-06  8.53014e-06

$hessian
          [,1]       [,2]
[1,] 8.4991797  0.4600109
[2,] 0.4600109 10.8685235

$code
[1] 1

$iterations
[1] 6

> out3 <- nlm(mlogl3, theta.start, hessian = TRUE, x = x)
> solve(out3$hessian)  ### asymptotic variance of MLE
             [,1]         [,2]
[1,]  0.117928566 -0.004991333
[2,] -0.004991333  0.092220076
> sqrt(diag(solve(out3$hessian)))  ### asymptotic s. d. of MLE
[1] 0.3434073 0.3036776
> out$estimate
Error: Object "out" not found
> out3$estimate
[1] -0.3856592  1.2444707
>