# How to find a maximum likelihood estimator?

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 :).

• 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 $x$, $f(x)=0$. – whuber Nov 3 '14 at 20:00
• Well I can only get to the answer that the hint says is wrong haha :) I dont understand where I am going wrong? – Joshdn10 Nov 6 '14 at 21:30
• 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? – Joshdn10 Nov 6 '14 at 21:34
• 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. – whuber Nov 7 '14 at 16:22