Questions about parameter estimation for the Cauchy Distribution.

$X_{1} … X_{n}$ are i.i.d. according to the Cauchy distribution centered at θ. The the density function is given by:

$frac{1}{pi} cdotfrac{1}{[1 + (x-theta)^2]}$

and the cdf is given by:

$frac{1}{2} + frac{1}{pi}arctan(x – theta)$

Find the estimator of $theta$. Using a concept similar to the method of moments, $hat p_{n}$ denotes the proportion of negative observations which is close to its expected value of $F_{theta}(0)$. Demonstrate that the estimator is consistent, and find the limiting distribution.

Couple questions here though:

1. For determining the estimator we set $hat p_{n}$ equal to its expected value of $F_{theta}(0)$ and solve for theta, correct? Here I obtained tan($frac{pi}{2} – pihat p)$ = $hat theta$

2. How do we demonstrate the consistency of the estimator though? Can we argue that $X_1,ldots,X_nstackrel{iid}{sim}mathrm{Ber}(p)$ because p is a proportion of negative observations and there can only be two outcomes. Thus $hat p_{n}$ = $frac{sumlimits_{i=1}^n X_i}{n}$ which means it converges to p using the weak law of large numbers? And because $hat p_n$ converges to p, $hat theta$ must converge to $theta$ as well due to the continuous mapping theorem?

3. I believe we can find the asymptotic distribution using the delta method. Is this correct as well?

Thanks, wondering here if I am on the right track. I don’t know if I am off base here, to be honest.

set.seed(111)
h = replicate(10^5, median(rt(100,1)))
summary(h);  sd(h)

      Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
-0.7395925 -0.1059586  0.0005111 -0.0000415  0.1050140  0.7057973 
[1] 0.1585002


a.38 = replicate(10^5, mean(rt(100,1),trim=.38))
summary(a.38);  sd(a.38)

      Min.    1st Qu.     Median       Mean     3rd Qu.       Max. 
-0.7270404 -0.1025867 -0.0009389 -0.0006116  0.1006515  0.7054522 
 [1] 0.1530064  # slightly smaller SD