# Pivot Quantity for Exponential Distribution

• October 24th 2012, 12:12 AM
DavidEriksson
Pivot Quantity for Exponential Distribution
Hi,

I am given two samples $\{x_1,x_2,...,x_n\} \sim \mbox{Exponential}(\lambda_1)$ and $\{y_1,y_2,...,y_m\} \sim \mbox{Exponential}(\lambda_2)$ and I wish to use a pivot quantity to test the hypothesis $H_0: \lambda_1=\lambda_2$ against $H_a: \lambda_1 \neq \lambda_2$ using a suitable pivot quantity.

I know that this means that I should find a statistics with a distribution that doesn't depend on $\lambda_1=\lambda_2$ under $H_0$. Since
$\sum_{i=0}^n X_i \sim Gamma(n,\lambda_1)$ this means that $\frac{\lambda_1}{n}\sum_{i=0}^n X_i \sim Gamma(n,n)$ and hence
T= $\frac{\lambda_1}{n}\sum_{i=0}^n X_i+\frac{\lambda_2}{m}\frac{m}{n}\sum_{i=0}^m Y_i \sim Gamma(m+n,n)$ is a quantity with a distribution that doesn't depend on $\lambda_1=\lambda_2$.

My questions are
1) is the above correct
2) How do I now proceed to compute a p-value. I see how I can simulate from an exponential distribution with any given parameter and compute the test statistics, but how do I evaluate the statistics for my given sample since I don't know $\lambda_1,\lambda_2$?