Can u help me?
consider a sample $X_1,X_2,\dots,X_n$ from the following density function
$$f_{\theta}(x)= \frac{1}{\theta}\exp\left[-\frac{1}{\theta}x\right],\quad x>0$$ where $\theta>0$ is an unknown parameter.
Show that the following estimator is weakly consistent for $\theta$:
$$T_n=\left(\frac{1}{n-1}\right)\sum_{i}X_i-\frac{X_1}{n}$$
$$ T_n = \frac{1}{n-1} \sum X_i - \frac{X_1}{n} = \left(\frac{n}{n-1}\right) \frac{1}{n}\sum X_i - \frac{X_1}{n}, $$ Using the WLLN $$ \frac{1}{n}\sum X_i \xrightarrow{p} EX = \theta, $$ and $n/(n-1) \to 1$ and $X_1/n \to 0$, hence $$ T_n \xrightarrow{p}1\times\theta+0=\theta. $$