# Thread: Constructing one-sided CI from MLE

1. ## Constructing one-sided CI from MLE

$\displaystyle X_1,X_2,...,X_n$ is a sample from the distribution whose density is:

$\displaystyle f_{X}(x) = e^{-x-\theta} \mbox{ if }x \geq \theta$

Based on the MLE estimator of $\displaystyle \theta$ construct a one-sided confidence interval for the unknown parameter at confidence level $\displaystyle 1-\alpha$

Here's what I have so far.

$\displaystyle f(x_1,...,x_n|\theta) = e^{-x_{i} + \theta}\cdot\cdot\cdot e^{-x_{n} + \theta}$

$\displaystyle =e^{\sum_{1}^{n} -x_i + n\theta}$

$\displaystyle \log{f(x_1...x_n|\theta)}=\sum_{i=1}^{n}-x_i + n\theta$
$\displaystyle \frac{d}{d\theta}f(x_1...x_n|\theta) = n$

$\displaystyle n=0 ?$

I'm not too sure where to go from here. Any help? Thanks.

2. YOU are ignoring the indicator function
BESIDES TWO TYPOs (the i and the negative sign in front of theta)
THE MLE is the smallestorder stat.
YOU need to use common sense, not calculus (incorrectly)
to maximize the likelihood function

$\displaystyle f(x_1,...,x_n|\theta) = e^{-x_{1} + \theta}\cdot\cdot\cdot e^{-x_{n} + \theta}I(X_{(1)}>\theta)$

$\displaystyle = e^{-\sum x_{i} + n\theta}I(X_{(1)}>\theta)$

This is smallest, ZERO, when $\displaystyle X_{(1)}<\theta$

while you want $\displaystyle e^{-\sum x_{i} + n\theta}I(X_{(1)}>\theta)$ as large as possible WITH RESPECT TO THETA

So, you will need $\displaystyle X_{(1)}>\theta$

so to make $\displaystyle e^{-\sum x_{i} + n\theta}I(X_{(1)}>\theta)$ as large as possible
you want $\displaystyle n\theta$ or $\displaystyle \theta$ as big as possible
and it CANNOT exceed the data, hence the MLE (is our sufficient statistic by the way) $\displaystyle X_{(1)}$

3. Thank you! I was wondering what X_{(1)} means? Is that just X_1?

Also, what is the sufficient statistic? I looked it up on wikipedia but I don't really understand it.

4. Originally Posted by BERRY
Thank you! I was wondering what X_{(1)} means? Is that just X_1?

Also, what is the sufficient statistic? I looked it up on wikipedia but I don't really understand it.

$\displaystyle X_{(1)}$ is the minimum of $\displaystyle X_1, X_2,...,X_n$