Alright so I have been working with this one for a while and i'm not totally sure where to take it
Let $X_1,.....X_n$ be a sample from a distribution with CDF
$F(x;\theta)= 1-C/((x-\theta)+3)^4 for$ $x > \theta, \theta > 0$
a) Determine the constant C of $F(x;\theta)$ and its density $f(x;\theta)$
b) Determine the MLE of $\theta$. Present convincing arguments to support your answer
c) Determine the CDF of the MLE and find the limit of it's distribution when n increase indefinitely
d) If the MLE consistent for estimating $\theta$? Prove your assertion.
e) What is the MME of $\theta$? Is it consistent for estimating $\theta$? Prove your assertion.
f) Use the CLT to obtain the $\sqrt n$ asymptotic distribution of the distribution MME($\theta$) with it's exact parameters
What I have so far is I believe the CDF is Pareto $\sim (\alpha=4, C=1)$ thus for a) i know the constant is 1 and then by differentiating wrt x I get:
$$f(x;\theta)=\frac{4}{((x-\theta)+3)^5}$$
I dunno I feel like I'm doing something wrong here and if I don't do these step right it will mess up the MLE in the next step.
ok so does this make sense?
$L(\theta)=\prod_{i=1}^n \frac{4}{(x-\theta+3)^5}$
$L(\theta)=\frac{4^n}{\prod_{i=1}^n(x-\theta+3)^5}$
$L(\theta)=nlog(4)- \prod_{i=1}^n 5 log((x-\theta)+3)$
Any guidance would be appreciated