91Ó°ÊÓ

The joint probability density function (pdf) of \(X_{1}, X_{2}, \ldots, X_{n}\) is given by \( f(x_{1}, x_{2}, \ldots, x_{n} ; \theta) = \prod_{i=1}^{n} f(x_{i} ; \theta)\). Since the \(X_{i}\)s are independent and identically distributed (iid), the joint pdf becomes: \(f(x_{1}, x_{2}, \ldots, x_{n} ; \theta) = \prod_{i=1}^{n} \frac{1}{3 \theta} = \left(\frac{1}{3 \theta}\right)^n\). The MLE \(\hat{\theta}\) is found by setting the derivative of the log-likelihood function to zero and solving for \(\theta\). The log of the likelihood function is \( L(\theta ) = \ln \left( \left(\frac{1}{3 \theta}\right)^n \right) = -n\ln (3\theta) \). On differentiating, we get \( L'(\theta ) = -\frac{n}{\theta} \). Setting this to zero, we find no solution. But observing the restriction -\(\theta < X < 2\theta\) , MLE \(\hat{\theta}\) is \(\frac{X_{(n)}}{2}\) where \(X_{(n)}\) is the largest observation.

The MLE \(\hat{\theta}\) is a function of the sample (specifically the largest observation). According to the Factorization theorem, if the joint pdf/pmf can be factored as a function of the parameters and a function of the statistic, then that statistic is sufficient. Here, since our MLE depends only on \(X_{(n)}\), \(\widehat{\theta}\) is a sufficient statistic for \(\theta\).

An estimator \(T\) is said to be MVUE only if \(T\) is unbiased and it has the smallest variance among all unbiased estimators for that parameter. Unbiasedness of \(\left( (n+1)\hat{\theta}/n \right)\) can be established as the expectation of this expression is \(\theta\) itself. However, without information regarding all other possible unbiased estimators of \(\theta\) and their variances, definitively stating that this estimator is the unique MVUE is not possible.