Asymptotic confidence interval for gamma distribution

Question

Let $X_1,...,X_n$ be a sample from a gamma distribution with parameters $(θ,λ)$. I need to find an asymptotic confidence interval for $θ$ confidence level $α$, When $λ$ is unknown. I don't understand how to get close to this problem, because when constructing an asymptotic confidence interval, you first need to find an asymptotic estimate for the parameter $\theta$: $$ \sqrt{n}\frac{(\frac{1}{n}\sum_{i=1}^nX_i - \theta\lambda)}{\sqrt{\theta\lambda}} \xrightarrow{d} N(0, 1) $$ if $\lambda$ was known, then we could consider $\frac{\frac{1}{n}\sum_{i=1}^nX_i}{\lambda}$ and it will be asymptotic estimate of $\theta$. But what to do in case, when $\lambda$ is unknown? Thank you for help!

Answer 1

According to comment of @Henry we could consider: $$ \sqrt{n}\frac{(\frac{\frac{1}{n}\sum_{i=1}^nX_i}{\lambda} - \theta)}{\sqrt{\theta\lambda}} \xrightarrow{d} N(0, 1) $$ then, because $\frac{1}{n}\sum_{i=1}^nX_i$ is a consistent estimator of $\theta\lambda$ and sample variance is a consistent estimator of $\theta\lambda^2$. We could write: $$ \sqrt{n}\frac{(\frac{(\frac{1}{n}\sum_{i=1}^nX_i)^2}{S^2} - \theta)}{\sqrt{\frac{1}{n}\sum_{i=1}^nX_i}} \xrightarrow{d} N(0, 1) $$ Out of here it's not difficult to find confidence interval.