MLE of Cauchy distribution in R I am trying to compute the following "approximate Maximum Likelihood Estimate" in R. I am a little lost as to how to do this though:

any hints would be appreciated.           
 A: You want to use the function nlm() to calculate the argmax, which requires a function to be optimized as the first argument and the an initial guess at the parameters as the second argument, like so:
# define g(x;gamma)
g  <-  function(x,gamma) {
    (1/pi) * ( gamma / (x^2 + gamma^2) )
}

# define y here so it can be used inside fun() via lexical scoping
y <- blah blah blah

# the function to be optimized is optimized over the first argument.
fun  <- function(x){
    # separate the first argument into the variables needed 
    # for opimization
    phi = x[1:2]
    psi = x[3:4]
    gamma = x[5]

    # PERFORM THE BIG SUM
    out  <-  0
    for(t in seq(r + 1,T-s)){
        # calculate A
        A  <-  (y[t]   * (psi[1]*phi[1] + 1)
                + (psi[1]*phi[2] + phi[1]) * y[t-1]
                - phi[2]* y[t-2]
                - psi[1] * y[t+1]
                )
        # increment the sum
        out  <-  out + log(g(A,gamma))
    }
    #nlm performs the argmin, so we'll return -out
    -out
}

# do the optimization
out  <-  nlm(fun,
        # initial guess at the parameters,
             c(0,0,# phi
               0,0,# psi
               0))# gamma

phi_hat   <- out$estimate[1:2]
 psi_hat   <- out$estimate[3:4]
gamma_hat <- out$estimate[5]

Note that nlm() does not make restrictions on the arguments so, for
example, if you need the arguments to be positive, you need to make a
transformation like this:
phi = exp(x[1:2])

in fun()  and also here:
phi_hat   <- exp(out$estimate[1:2])

