# Maximum likelihood estimators of $\theta$ in $U(2\theta-1,2\theta+1)$ distribution

I understand why (D) is one of the answers but i dont know about the rest?

• One approach would be to compute the likelihoods for all four estimators. Another is to compute the maximum likelihood estimate outright. Have you tried either of these? – whuber Feb 8 at 17:43
• Please add the 'self-study' tag and read the tag wiki. – StubbornAtom Feb 8 at 18:17
• If any statistic $T$ such that $T_1<T<T_2$ is an MLE, then $\alpha T_1+(1-\alpha)T_2$ is also an MLE for any $\alpha\in(0,1)$ since it lies between $T_1$ and $T_2$. As option (D) is correct, check if you can write the estimators in options (A),(B),(C) in the form $\alpha(\frac{Y_n-1}{2})+(1-\alpha)(\frac{Y_1+1}{2})$. – StubbornAtom Feb 11 at 9:46