asymptotic confidence interval of poisson distribution

Question

Find the variance stabilizing transformation $g$ that satisfies $$\sqrt{n}(g(\bar{X_{n}})-g(\lambda)) \rightarrow Z, Z \sim N(0,1)$$ where $\bar{X_{n}}$ is a sample mean of random variables $X_{1}, \cdots, X_{n}$ that follow Poisson distribution $\operatorname {Poisson}(\lambda)$. Use the result to find the asymptotic confidence interval in terms of $\lambda$.

First, $$\sqrt{n}(\bar{X_{n}}-\lambda) \rightarrow N(0,\lambda)$$ holds from CLT. Then, $$\sqrt{n}(g(\bar{X_{n}})-g(\lambda)) \rightarrow N(0,\{g'(\lambda)\}^{2}\lambda)$$ , which should be equivalent to $N(0,1)$, so $g'(\lambda) = \frac{1}{\sqrt{\lambda}}$ and $g(\lambda) = 2\sqrt{\lambda}$. Therefore, $$\sqrt{n}(\sqrt{\bar{X_{n}}}-\sqrt{\lambda}) \rightarrow N(0,\frac{1}{4}).$$ Now how should I proceed further?

Answer 1

You are almost there!

You correctly found that the "variance stabilizing transformation $g()$" is $2\sqrt{()}$

that is

$$\sqrt{n}\left[2\sqrt{\overline{X}_n} -2\sqrt{\lambda} \right]\dot{\sim}N(0;1)$$

Now all you have to do is to fix the two Gaussian quantiles with a certain significance level $\alpha$, set

$$-z<\sqrt{n}\left[2\sqrt{\overline{X}_n} -2\sqrt{\lambda} \right]<z$$

and solve the double inequality in $\lambda$