# confidence interval of $\beta$, where $X$'s are from exponential distribution

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 $$\beta$$?

• MLE is probably asymptotically normal using $\sqrt n(\hat\beta-\beta)\stackrel{d}\to N(0,1/I(\beta))$. Mar 22, 2021 at 14:27
• Thanks. But I am not sure if CLT can be useful here.
– Tan
Mar 23, 2021 at 16:34
• I was referring to the asymptotic normality of MLE under quite general conditions. Mar 23, 2021 at 20:22

I am answering my own question. Please correct me if I make any mistakes.

Continuing with $$T=n[-\log(\hat{\beta})+\log(\beta) - 1 +\frac{\hat{\beta}}{\beta}],$$ we can write $$T=n[\log\frac{\hat{\beta}}{\beta} - 1 +\frac{\hat{\beta}}{\beta}]$$

Now, let $$h(x)=\log(x)-1+x,$$ and we see $$h(x)$$ achieves the minimum when $$x=1$$. This means $$T$$ is the smallest when $$\frac{\hat{\beta}}{\beta}=1$$.

In order words, it is equivalent to accept the null hypothesis ($$\beta=\beta_0$$) when $$T$$ is small or when $$|\frac{\hat{\beta}}{\beta}-1|$$ is small.

Thus, the confidence interval of $$\beta$$ is simply

$$-c<\frac{\hat{\beta}}{\beta}-1 which gives the confidence interval of $$\beta$$.

• You will need to define $c$ in terms of the known quantities. Furthermore, there should probably be no $\beta_0$ - there is no null hypothesis behind the concept of a confidence interval. Mar 23, 2021 at 16:51