# Approximate distribution involving a log. Still an easy CLT application?

• Nov 19th 2011, 10:01 PM
subfallen
Approximate distribution involving a log. Still an easy CLT application?
Fix $p \in (0,1)$. Suppose $P_{n}$ has distribution $\text{Bin}(n,p)$. Set $\hat{P_{n}} := \frac{P_{n}}{n}}$. What is an asymptotic distribution for...

$\log\frac{\hat{p_{n}}}{1-\hat{p_{n}}}$

...?
• Nov 19th 2011, 10:45 PM
CaptainBlack
Re: Approximate distribution involving a log. Still an easy CLT application?
Quote:

Originally Posted by subfallen
Fix $p \in (0,1)$. Suppose $P_{n}$ has distribution $\text{Bin}(n,p)$. Set $\hat{P_{n}} := \frac{P_{n}}{n}}$. What is an asymptotic distribution for...

$\log\frac{\hat{p_{n}}}{1-\hat{p_{n}}}$

...?

I would put: $\widehat{p}_n=p+\epsilon$, where $\epsilon \sim N(0,{p(1-p)/n})$

Then expand the $\log$ as a power series in $\epsilon$ and proceed from there (you will end up ignoring all but the first term in $\epsilon$, or all but the first two terms if you want to estimate the bias correction).

Spoiler:

I include this as I will eventually lose my notes:

asymtotically

$\log \left( \frac{\hat{P_n}}{1-\hat{P_n}} \right) \sim N \left( \log \left(\frac{p}{1-p}\right), \frac{1}{np(1-p)} \right)$

Code:

>
>p=0.25;
>n=10000;
>sigma=sqrt(p*(1-p)/n);
>
>phat=normal(1,100000)*sigma+p;
>
>logp=log(phat/(1-phat));
>histogram(logp);
>
>{m,s}=meandev(logp);[m,s]
-1.09861    0.0231407
>
>log(p/(1-p))
-1.09861
>sqrt(1/(n*p*(1-p)))
0.023094
>
>
>BiasBound=(2*p-1)*p*(1-p)/(2*p^4-4*p^3+2*p)/n
-2.10526e-005
>

taylor(log((p+%epsilon)/(1-p-%epsilon)), %epsilon, 0, 3);

$-\mathrm{log}\left( p-1\right) +\mathrm{log}\left( p\right) +\mathrm{log}\left( -1\right) -\frac{\epsilon}{{p}^{2}-p}+\frac{\left( 2\,p-1\right) \,{\epsilon}^{2}}{2\,{p}^{4}-4\,{p}^{3}+2\,{p}^{2}}-\frac{\left( 3\,{p}^{2}-3\,p+1\right) \,{\epsilon}^{3}}{3\,{p}^{6}-9\,{p}^{5}+9\,{p}^{4}-3\,{p}^{3}}+...\]$

CB
• Nov 21st 2011, 12:39 PM
subfallen
Re: Approximate distribution involving a log. Still an easy CLT application?
CaptainBlack -

Very clever! I eventually realized that this can be handled more directly by the delta method, fwiw.

Also, you wanted the mean of the approximate distribution to be $\log\frac{p}{1-p}$ instead of $\frac{p}{1-p}$, right?

Yes.

CB