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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 $H_{a}: \lambda <k\mu$ 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.

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