I'm probably really overcomplicating things but I want to specify the likelihood ratio test with significance level $\alpha = 0.05$
I have three random samples (sample sizes: $n_1, n_2$ and $n_3$), with sample means $X$ and $Y$ and $Z$, respectively. $X_i$ has an $\exp(\mu_1)$ distribution, $Y_i$ has an $\exp(\mu_2)$ distribution, and $Z_i$ has an $\exp(\mu_3)$ distribution.
I understand how to do it for one random sample but I don't understand how to approach it for three samples.
MLE of $\lambda$ is the reciprocal of the sample mean.
H0 :$\mu_1=\mu_2 =\mu_3$ versus H1 : H0 is not true.
If $X$, $Y$, and $Z$ are independent and have the same mean, then the sum $T=X+Y+Z$ is Gamma distributed with $\alpha=3$ and $\beta=\lambda$. So, under the zero hypothesis, the sum is a Gamma random variable; then, you use a test (for instance, a qq plot or a histogram) to see if the data match a Gamma random variable with the required confidence.