Results 1 to 3 of 3

Math Help - Approximate distribution involving a log. Still an easy CLT application?

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    11

    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}}}

    ...?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

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

    Quote Originally Posted by subfallen View Post
    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
    Last edited by CaptainBlack; November 21st 2011 at 08:30 PM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    11

    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
    Last edited by CaptainBlack; November 21st 2011 at 08:28 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Approximate with a Normal distribution (Bayesian statistics)
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 4th 2010, 07:42 AM
  2. findings mean, SD and approximate distribution
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: June 30th 2009, 10:28 AM
  3. normal distribution to approximate
    Posted in the Advanced Statistics Forum
    Replies: 4
    Last Post: June 1st 2009, 09:09 PM
  4. Application involving matrices!!!
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 23rd 2007, 01:39 AM
  5. Replies: 1
    Last Post: March 27th 2007, 04:50 PM

Search Tags


/mathhelpforum @mathhelpforum