Confidence Interval for a function of a parameter

[alexhleb366]

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/Binomial_proportion_confidence_interval

[statdad]

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

For a big sample size \hat p has approximately a normal distribution, right? You can approximate the distribution of \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 p^2

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

If an estimate X_n for some parameter \theta satisfies


\sqrt n \left(X_n - \theta \right) \sim n(0, \sigma^2)


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

then for a function f that is continuous and has a non-zero derivative at \theta it is true that


\sqrt{n} \left(f(X_n) - f(\theta)\right) \sim n(0, \sigma^2 f'(\theta) \right)


Your statistic is the sample proportion, the parameter is p , and the function is f(x) = x^2