Question

A random sample of size n is drawn from a uniform population over (θ-1/3,θ+1/3),Obtain maximum likelihood estimator ofθ.

Answer 1

likelihood function for a uniform distribution:

$L(\theta)=\Pi^n_{i=1}f(x_i;a,b)=\Pi^n_{i=1}\frac{1}{(b-a)^n}=\Pi^n_{i=1}\frac{1}{(b-a)^n}=\Pi^n_{i=1}\frac{1}{(\theta +1/3-\theta +1/3)^n}=\Pi^n_{i=1}\frac{1}{(2/3)^n}$

log-likelihood function:

$l(\theta)=-nlog(2/3)$

log-likelihood function does not depend on $\theta$ , so MLE for $\theta$ does not exist