Consider $f_θ(x) = \exp(θ − x)\, \mathbb{I}_{x>θ}$ show that $\inf(X)$ is complete and sufficient

Question

Consider a sequence of variable defined with the following density $$ f_θ(x) = \exp(θ − x)\ \mathbb{I}_{x>θ} $$

1. What is the ML estimate?

2. Show that some statistics of the indicator functions $\mathbb{I}_{x_i>θ}$ are sufficient and complete by deducing an unbiased estimator of $θ$

3. For $θ^2$ what could an unbiased estimator be?

I see that the model is not regular and that MLE does not exist, how to find another unbiased estimator for $θ$ and $θ^2$?

Answer 1

Indications to help you towards solving this exercise:

1. If you observe iid rv's from this exponential distribution,$$x_1,\ldots,x_n\stackrel{\text{iid}}{\sim} f_\theta(x)$$can you write down the likelihood$$\begin{align*} L(\theta|x_1,\ldots,x_n)&=\prod_{i=1}^n f_\theta(x_i)=\prod_{i=1}^n e^{\theta-x_i}\mathbb{I}_{x_i>\theta}\\&=\exp\{n\theta-\sum_{i=1}^nx_i\}\mathbb{I}_{\inf\{x_i\}>\theta}\end{align*}$$and deduce the MLE?
2. When considering $S(x_1,\ldots,x_n)=\inf \{x_1,\ldots,x_n\}$ can you derive the probability density of $S$ and show that the density of $x_1,\ldots,x_n$ only depends on $S$ modulo a multiplicative constant? Can you find a moment of $S$ that is a linear function of $\theta$?
3. Can you express $\theta^2$ as a moment of $S$?