Suppose $X_i\overset{ind}{\sim}\mathcal{E}(\lambda_i)$, where $\lambda_i=(t_i\beta)^{-1}$, where $t_i$'s are positive known values and $\beta$ is positive unknown parameter. Here $i=1,\dots,n$.
It can be calculated that the MLE of $\beta$, $$\hat{\beta}_{MLE}=\frac{1}{n}\sum_{i=1}^n\frac{X_i}{t_i}$$
What is the confidence interval of $\beta$?
I have tried the likelihood ratio test, which gives the test statistic, namely $$\frac{p(\hat{\beta})}{p(\beta_0)}=\left(\frac{\hat{\beta}}{\beta_0}\right)^n\exp\left\{\left[\frac{1}{\beta_0}-\frac{1}{\hat{\beta}}\right]\sum_{i}^n\frac{X_i}{t_i}\right\},$$ and by taking log we get $$n[-\log(\hat{\beta})+\log(\beta_0)-1+\frac{\hat{\beta}}{\beta_0}].$$ I was trying to use the central limit theorem and delta method, but it seems not working. What can we do to prove that we can get the confidence interval for $\beta$?