Confidence Interval for a function of a parameter

**alexhleb366** · May24-09, 12:20 AM

How would you find CI for some function of a parameter?

the case i'm particularly interested in is an approximate CI for p^2
where p is the proportion in the binomial model.

http://en.wikipedia.org/wiki/Binomia...dence_interval

**statdad** · May25-09, 03:32 PM

I'm not sure of your math-stat background in this problem, so bear with me.

For a big sample size $LaTeX Code: \\hat p$ has approximately a normal distribution, right? You can approximate the distribution of $LaTeX Code: \\hat{p}^2$ (it will also turn out to be a normal distribution - look in (say) Hogg/Craig or any introductory math stat book for the idea, or write back and I can put the method here), and then you can get an approximate confidence interval for

Note - just so I don't have to post it:

If an estimate

for some parameter $LaTeX Code: \\theta$ satisfies

$LaTeX Code: <BR>\\sqrt n \\left(X_n - \\theta \\right) \\sim n(0, \\sigma^2)<BR>$

(the $LaTeX Code: \\sim$ means "tends to a normal distribution as $LaTeX Code: n \\to \\infty$ - i.e., it represents convergence in distribution)

then for a function

that is continuous and has a non-zero derivative at $LaTeX Code: \\theta$ it is true that

$LaTeX Code: <BR>\\sqrt{n} \\left(f(X_n) - f(\\theta)\\right) \\sim n(0, \\sigma^2 fsingle-quote(\\theta) \\right)<BR>$

Your statistic is the sample proportion, the parameter is

, and the function is