# Exact/Approximate Confidence Interval of Parameter Ratio from two Samples of iid Exponential

Suppose 2 independent samples $$X_1,...,X_n \sim Exp(\lambda_1)$$ and $$Y_1,...,Y_m \sim Exp(\lambda_2)$$, and are iid within samples.

I am thinking about how to make an exact confidence interval for $$\theta$$ where $$\theta = \lambda_1/\lambda_2$$. Some immediate thoughts, I can try to come up with a pivotal quantity, such as $$\lambda_1 \sum X_i \sim Gamma(n,1)$$ and $$\lambda_2 \sum Y_i \sim Gamma(m,1)$$. If I take the ratio of the two, I have $$\frac{m \lambda_1 \sum X_i}{n\lambda_2 \sum Y_i} \sim F_{2n,2m}$$.

In light of the duality between confidence intervals and hypothesis testing, say we wish to test $$H_0: \theta \equiv \frac{\lambda_1}{\lambda_2} = \theta_0$$ and $$H_a: \theta \equiv \frac{\lambda_1}{\lambda_2} \neq \theta_0$$. Then from above, we can use the fact that $$\frac{m \lambda_1 \sum X_i}{n\lambda_2 \sum Y_i} \sim F_{2n,2m}$$ to get $$\frac{\sum X_i}{\sum Y_i}\sim \frac{n}{m\theta_0} F_{2n, 2m}$$. So we would reject $$H_0$$ at level $$\alpha$$ by setting the rejection region as $$\frac{\sum X_i}{\sum Y_i} \leq \frac{n}{\theta_0 m}F_{2n, 2m}(\alpha/2)$$ and $$\frac{\sum X_i}{\sum Y_i} \geq \frac{n}{\theta_0 m}F_{2n, 2m}(1-\alpha/2)$$.

Thus the confidence interval, in which we are more concerned with the complement of the rejection region over $$\theta$$, would be $$\frac{n\sum Y}{m \sum X}F_{2n, 2m}(\alpha/2) \leq \theta \leq \frac{n\sum Y}{m\sum X}F_{2n, 2m}(1-\alpha/2)$$.

To test this theory out, I used R to simulate 23 observations from $$Exp(\lambda_1= 3)$$ and 45 observations from $$Exp(\lambda_2 = 5)$$. $$\sum X = 4.2689$$ and $$\sum Y = 6.8285$$. We know that the true $$\theta = \lambda_1/\lambda_2 = 0.6$$. Using the confidence interval built above, we get $$\theta \in (0.4819, 1.3298)$$ at the $$\alpha = 0.05$$ level.

Are there other ways to make an exact test that leads to different intervals and rejection regions?

How would one go about making an approximate/large sample test for this?

Note: This is not from a problem set. I am just completing some thought exercises based off statistics courses I taken many years ago.