Confidence interval for parameter

Question

$X_1, X_2, \dots, X_n$ - i.i.d observations

$X_1 = \xi + \eta$ where $\xi \sim N(\theta^2, \theta^2+1)$, $\eta = \begin{cases} 0, & 1/2 \\ 4\theta, & 1/2 \end{cases}$, $\xi$ and $\eta$ are independent.

Find $\alpha$-confidence interval.

The first that I need to do is to find some estimate for $\theta$. The only one that I find is $(5\overline{X} - S^2 +1)/10$ but it is difficult to find distribution.

Is it possible to do something else?

Answer 1

A natural choice arises from the method of moments estimator: $$\bar X = \operatorname{E}[\xi + \eta] = \theta^2 + 2\theta,$$ thus if $\theta > 0$, $$\hat \theta = \sqrt{1+\bar X} - 1.$$ Then, what is the variance of $\hat \theta$? You could use the delta method: $$\operatorname{Var}[g(X)] \approx (g'(\operatorname{E}[X]))^2 \operatorname{Var}[X].$$ This will give you a Wald-type asymptotic confidence interval.