Re: Median likelihood?
- From: FD <wentriken@xxxxxxxxxxx>
- Date: Thu, 27 Aug 2009 09:51:55 -0700 (PDT)
On Aug 13, 2:56 pm, Ken Butler <but...@xxxxxxxxxxxxxxxx> wrote:
On Thu, 13 Aug 2009 09:45:53 -0700 (PDT), FD <wentri...@xxxxxxxxxxx>
wrote:
For a bernoulli trial, is there a significant meaning for the median
of the likelihood function or the median of the log likelihood
function?
Median is of a probability distribution (such as the sampling
distribution of a statistic), not of a (log) likelihood. So you'll
need to be clearer in your definition.
Example:
Observed: HHT (s=2;f=1)
Median likelihood: 0.61
Maybe "median unbiased estimation" is what you want. In the same way
that an estimator is called unbiased if the mean of its sampling
distribution is equal to the parameter being estimated, an estimator
is called median unbiased if the *median* of its sampling distribution
is equal to the parameter being estimated.
The only context I can see in which what you're doing makes any sense
is a Bayesian one. Let P(success)=p; then p in Bayesian terms is a
random variable whose posterior density is its prior density times the
likelihood. If the prior is constant (that is, p has a prior uniform
distribution), the posterior density is the same as the likelihood.
In your case, likelihood times prior is (proportional to) p^2(1-p)
(times 1), and the posterior median can be found by equating the
integral of this from 0 to m (wrt p) to 0.5, and solving for m. If you
are looking for an estimator of p with minimum absolute error loss, I
guess this is what you'd do.
--
Ken Butler, Lecturer (Statistics)
University of Toronto at Scarborough
butler (at) utsc.utoronto.cahttp://www.utsc.utoronto.ca/~butler
Thank you, this is very informative and yes I am using a bayesian
application. Could you please tell me a few keywords to start my
research, specifically what the p^2(1-p) is called? It is not clear to
me how to take the sampled number of successes and failures with that
formula.
.
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