Maximum Likelihood Estimator for a exp(1/$\theta$) distributed rv

Let $$n \in \mathbb{N}, \, X_i: (0,\infty)^n \rightarrow (0,\infty)$$ so that $$X_i(x_1,...,n_n) = x_i \, \forall i \in \{ 1,...,n \}.$$ Let $$((0,\infty)^n, B((0,\infty)^n), (P_\theta)_{\theta \in (0,\infty)})$$ be a statistical modell, so that $$(X_1,....,X_n)$$ is under $$P_\theta$$ a iid family of $$\exp(\frac{1}{\theta})$$ distributed rv´s.

Note that in this case $$\forall i \in\{ 1,...,n \}$$ and $$\theta \in (0,\infty)$$ it holds that $$Var(X_i) = \theta^2$$.

a) Compute $$\forall x\in (0,\infty)^n$$ the MLE $$\hat{\theta}_{ML} \in (0,\infty)$$ for $$\tau(\theta) = \theta.$$

b)Is $$\hat{\theta}_{ML}$$ a variance minimizing unbiased estimator for $$\tau(\theta) = \theta$$?

to a):

the density of a $$exp(\theta^{-1})$$ distributed rv is: f(x) = $$\theta^{-1} * e^{-\theta^{-1}x}$$.

I get the maximum likelihood function:

$$L(x,\theta) = \prod_{i=1}^n f(x)$$ $$= \theta^{-n} \prod_{i=1}^n e^{\theta^{-1}x_i}$$

Now i build the log likelihood function and get:

$$\log(L(x,\theta)) = -n\log(\theta) - \frac{1}{\theta} \sum_{i=1}^n x_i$$

I set the derivate of the $$\log$$ likelihood function $$=0$$ to find the maximum.

$$\Rightarrow -\frac{n}{\theta}+\frac{1}{\theta^2} \sum_{i=1}^{n} x_i =0$$

$$\Leftrightarrow \frac{n}{\theta} = \frac{1}{\theta^2} \sum_{i=1}^n x_i$$

$$\Leftrightarrow \theta = \frac{\sum_{i=1}^n x_i}{n}$$

so i get that my estimator is: $$\hat{\theta}_{ML} = \frac{\sum_{i=1}^n x_i}{n}$$

I don´t know why we get the hint that $$\theta^2 = Var(X_i)$$. Does someone see my mistake?

to b)

I know how to check that a estimator is unbiased. How do i check if a estimator is a variance minimizing unbiased estimator? Do i have to use Cramer - Rao - Inequality?

• Firstly note that $$\log\left(\prod_{i=1}^n e^{\theta^{-1}x_i} \right)=- \frac{\color{red}1}{\theta} \sum_{i=1}^n x_i$$ – callculus Feb 20 at 15:29
• thank you, i have improved my mistake. Is the result now correct? – Kaya Feb 20 at 15:48
• Yes, it is correct now. – callculus Feb 20 at 15:48
• would you mean the hint, that $\theta^2 = Var(X_i)$ is just for b) ? and do you have an idea how to show b) ? – Kaya Feb 20 at 15:50
• Yes, you can use Cramer - Rao Inequality. – NCh Feb 21 at 14:24