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-   -   Confidence interval (http://www.physicsforums.com/showthread.php?t=255541)

twoflower Sep11-08 03:02 PM

Confidence interval
 
Let's say we know this:

[tex]
\sqrt{n}\left(\widehat{\theta} - \theta\right) \sim \mathcal{N}\left(0, \frac{1}{F(\theta)}\right)
[/tex]

How do we get from this information to this expression of confidence interval for [itex]\theta[/itex]?

[tex]
\left( \widehat{\theta} \pm u_{1-\frac{\alpha}{2}}\frac{1}{\sqrt{nF\left(\widehat{\theta}\right)}}\right )
[/tex]

Where [itex]u_{1-\frac{\alpha}{2}}[/itex] is appropriate quantil of standard normal distribution.

Thank you.

statdad Sep11-08 06:02 PM

Re: Confidence interval
 
If [tex] a [/tex] is the value from [tex] Z [/tex] (standard normal) with area [tex] {\alpha}/2[/tex] to its right, you know the value of

[tex]
\Pr\left(-u < \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) < u)
[/tex]

because of your stated approximate normality result. That means the event

[tex]
-u < \sqrt{n F(\theta)} \left(\hat \theta - \theta\right) < u
[/tex]

has a known probability of occurring. What can you do with this inequality? (Try some work and include it with your next question if you are unsure.)


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