# Test composite hypothesis of 2 parameter $H_0: \lambda \ge k \mu$

I have a homework problem as follows: $$(X_i, Y_i), i = 1, \cdots, n$$ is a random sample from $$f(x,y|\lambda, \mu) = \lambda \mu e^{-\lambda x - \mu y}, x,y>0, \lambda, \mu>0$$. Test the hypothesis $$H_0: \lambda \ge k \mu$$ vs $$H_{a}: \lambda at level $$\alpha$$ where $$k$$ is some known constant.

My attempt: Let $$\theta_1 = \lambda - k\mu, \theta_2 = \lambda + k\mu$$, then let $$T_1 = k\Sigma X + \Sigma Y, T_2 = k \Sigma X - \Sigma Y$$, using transformation:

$$f(T_1, T_2|\theta_1, \theta_2) = c(\theta_1, \theta_2) \times \exp\{\frac{\theta_2}{2k} t_1 - \frac{\theta_1}{2k} t_2\} \times h^{*} (t_1 ,t_2)$$

My thinking is to find $$f(T_1|T_2)$$ for which I need $$f(T_2)$$ but I get stucked here. I also feel like there should be other easy approaches to this problem.

I'd appreciate any hints.

• Please add the self-study tag. Commented May 11, 2020 at 19:40
• Which test are you applying? If you try a likelihood ratio test, here is a related question: stats.stackexchange.com/q/81151/119261. Commented May 11, 2020 at 19:56