I include this as I will eventually lose my notes:
asymtotically
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);