# Likelihood Ratio Test of exp(λ) distribution with multiple samples

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.

• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For typesetting, please use MathJax. – dantopa May 24 '19 at 5:04
• The method is the same. Given the sample $(x_1,\ldots,x_{n_1},y_1,\ldots,y_{n_2},z_1,\ldots,z_{n_3})$ write down the likelihood function $L(\mu_1,\mu_2,\mu_3)=\prod_{i=1}^{n_1}f_{X_1}(x_i)\prod_{i=1}^{n_2}f_{Y_i}(y_i)\prod_{i=1}^{n_3}f_Z(z_i)$. Form the likelihood ratio statistic $\Lambda=\frac{\sup_{H_0}L(\mu_1,\mu_2,\mu_3)}{\sup_{H_o\cup H_1}L(\mu_1,\mu_2,\mu_3)}$ and remember to reject $H_0$ for small values of $\Lambda$. So start by finding the maximum likelihood estimate of $(\mu_1,\mu_2,\mu_3)$ under $H_0$ and under $H_0\cup H_1$. – StubbornAtom May 24 '19 at 19:24
• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, use MathJax. – dantopa May 25 '19 at 3:49

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.