Solution
= Solution
The log likelihood is
$$
\ell(\theta)=n\log\theta-\theta\sum_{i=1}^nX_i.
$$
Its strictly concave maximizer is
$$
\boxed{\widehat\theta_n=\frac{n}{\sum_iX_i}=\frac1{\overline X}.}
$$
= Solution
The log likelihood is
$$
\ell(\theta)=n\log\theta-\theta\sum_{i=1}^nX_i.
$$
Its strictly concave maximizer is
$$
\boxed{\widehat\theta_n=\frac{n}{\sum_iX_i}=\frac1{\overline X}.}
$$