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

  1. #1
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    Approximate distribution involving a log. Still an easy CLT application?

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

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

    ...?
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  2. #2
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    Re: Approximate distribution involving a log. Still an easy CLT application?

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

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

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

    Then expand the $\displaystyle \log$ as a power series in $\displaystyle \epsilon$ and proceed from there (you will end up ignoring all but the first term in $\displaystyle \epsilon$, 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

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

    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);

    $\displaystyle -\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
    Last edited by CaptainBlack; Nov 21st 2011 at 07:30 PM.
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  3. #3
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    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 $\displaystyle \log\frac{p}{1-p}$ instead of $\displaystyle \frac{p}{1-p}$, right?

    Yes.

    CB
    Last edited by CaptainBlack; Nov 21st 2011 at 07:28 PM.
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