Likelihood ratio test MLE Question

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For working out and maximizing the MLE of the general case, how do you work out the MLE for theta?

I can work out the MLE for beta when theta isn't 0. Is the MLE for theta obtained when theta = mean of xi? However, this causes my MLE for beta to = 0.

Any help would be greatly appreciated. Thanks!

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No MLE of theta is not mean of Xi . But it is MinXi because likelihood function is increasing function of theta and theta is maximum at Min Xi . so MLE of theta is MinXi. So MLE for beta is( summation Xi-nMinXi)/n