How to compute the consistency of an estimator

Question

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 $\theta$:

$$T_n=\left(\frac{1}{n-1}\right)\sum_{i}X_i-\frac{X_1}{n}$$

Here's a hint. Show that $T_n = \bar X + \frac{1}{n-1}\bar X - \frac{X_1}{n}$. Use the weak law of large numbers on $\bar X$, and show that the other two converge in probability to 0. — Matthew Towers, Dec 06 '17 at 12:19

score 0 · Answer 1 · answered Dec 08 '17 at 22:09

0

$$ T_n = \frac{1}{n-1} \sum X_i - \frac{X_1}{n} = \left(\frac{n}{n-1}\right) \frac{1}{n}\sum X_i - \frac{X_1}{n}, $$ 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 $$ T_n \xrightarrow{p}1\times\theta+0=\theta. $$

answered Dec 08 '17 at 22:09

V. Vancak

16,444

Thank you.. in this way I demonstrate that T is weakly consistent right? – Angela Francesca Papa Dec 10 '17 at 17:05
Yes. Weakly consistent = converges in probability. – V. Vancak Dec 10 '17 at 17:25

How to compute the consistency of an estimator

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