maximum likelihood estimator

Question

Suppose, $X_1,X_2,\ldots,X_n$ be a random sample from $U(\theta-2,\theta+2)$. Define, $X_{(n)}=\rm{max}\{X_1,X_2,\ldots,X_n\}$ and $X_{(1)}=\rm{min}\{X_1,X_2,\ldots,X_n\}$. Then which of the following estimators are MLE of $\theta$?

1. $X_{(1)}-2$;

2. $X_{(n)}+2$;

3. $\frac{X_{(1)}+X_{(n)}}{2}$;

4. $0.25(X_{(n)}-2)+0.75(X_{(1)}+2)$.

Ans: I have found by the method of MLE that any value lies within the interval $((X_{(n)}-2),(X_{(1)}+2))$ can be considered as MLE.

Then we can conclude that all the four values mentioned above are MLEs of $\theta$.

Is it correct??

Answer 1

Well no, because $$X_{(1)}-2<X_{(n)}-2$$ and therefore $$X_{(1)}-2\notin[X_{(n)}-2,X_{(1)}+2]$$

Furthermore $$X_{(n)}+2>X_{(1)}+2 \Rightarrow X_{(n)}+2\notin[X_{(n)}-2,X_{(1)}+2]$$