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Let $X_1,X_2$ be a random sample of size $n = 2$ from a distribution with density function given by:

$$f(x) = 2(x - q) , q \lt x \lt q + 1.$$

a) Show $E[(X - q)^k] = 2/(k+ 2)$ for $k \gt 0$.

b) Find the MME (method of moments estimate) for $q$.

c) Find the MLE (maximum likelihood estimate) for $q$. (The answer is not $(X_1 + X_2)/2$.)

The first part is no problem, and I get the sample mean $+ 2/3$ for the MME but struggling with part c) though. Any help is appreciated :).

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    $\begingroup$ Are you sure you don't mean 'sample mean -2/3' for the method-of-moments estimator (MME)? Exactly how are you "struggling"? Where are you stuck in maximizing the likelihood? BTW, the hint in (c) is to keep you from forgetting that the formula for $f$ applies only when $q\lt x\lt q+1$: for all other $x$, $f(x)=0$. $\endgroup$ – whuber Nov 3 '14 at 20:00
  • $\begingroup$ Well I can only get to the answer that the hint says is wrong haha :) I dont understand where I am going wrong? $\endgroup$ – Joshdn10 Nov 6 '14 at 21:30
  • $\begingroup$ Could you say that in order to maximise the likelihood we would need the smallest q possible and therefore the mle for q would be the first order stat of X? $\endgroup$ – Joshdn10 Nov 6 '14 at 21:34
  • $\begingroup$ Yes, I think you could say that. A similar example--and an illustration of one way to handle the restriction $q\lt x\lt q+1$ in the formulas--appears at stats.stackexchange.com/a/122856. $\endgroup$ – whuber Nov 7 '14 at 16:22
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So to start you find the likelihood function, so how do you go about this? We only have two data points so this simplifies the problem quite a bit. Once you find the likelihood then maximize the likelihood function. So where exactly are you stuck? What do you get as the likelihood function?

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  • $\begingroup$ My likelihood function gets me to the exact answer that the hint says is wrong. I'm not sure where I'm breaking some or other rule. $\endgroup$ – Joshdn10 Nov 6 '14 at 21:32
  • $\begingroup$ It could be that you do not have a unimodel likelihood, you need to check that. Plot the likelihood and see how many peaks there are from q to q+1 $\endgroup$ – Lauren Goodwin Nov 7 '14 at 20:14
  • $\begingroup$ Joshdn10, rather than telling us what you think the likelihood gives you, show us why you think it ... i.e. show how you got your answer, so we can explain your confusion. $\endgroup$ – Glen_b Dec 6 '14 at 22:54

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