Maximum likelihood estimator of $1/v$

Question

I am confused about the following question.

Let $X_1, \dots, X_n$ be i.i.d distributed with probability function \begin{equation} p_v(x) = \lambda \exp(-\lambda x) \end{equation} for $\lambda > 0$. Find the Maximum Likelihood Estimator for $1/ \lambda$.

I don't quite understand which of the following two approaches I should follow:

1. Compute $v^{*,1} = \arg\max_v L_X(v) = \prod_{i} v \exp(-v x_i)$ and then use $1/v^{*,1}$ as estimator.

2. Compute $v^{*,2} = \arg\max_v L_X(\frac{1}{v}) = \prod_{i} \frac{1}{v} \exp(-\frac{1}{v} x_i)$ and then use $v^{*,2}$ as estimator.

In this case, they seem to evaluate to the same result. However, I would really like to know which of these two approaches is the correct idea.

Thanks for your help!

Answer 1

They are equivalent due to the property of functional equivalence, i.e.

$$\widehat{g(\theta)}=g(\hat \theta)$$

where hats indicate ML estimators. Your setup is the case where $g:t\rightarrow1/t$.