Re: Mathematica Integration help
- To: mathgroup at smc.vnet.net
- Subject: [mg86177] Re: Mathematica Integration help
- From: Mark Fisher <particlefilter at gmail.com>
- Date: Wed, 5 Mar 2008 03:38:13 -0500 (EST)
- References: <fqetdm$jjs$1@smc.vnet.net> <fqghgr$f65$1@smc.vnet.net>
On Mar 4, 2:15 am, Francogrex <fra... at grex.org> wrote:
> On Mar 3, 10:48 am, antony.sea... at gmail.com wrote:
>
> > What you're integrating doesn't make any sense as a statistical
> > quantity. Did you intend to have different mu_i and integrate over
> > each such variable? At the moment you're averaging the likelihood of
> > {x_i} arising from Gaussians centred on the diagonal {mu, mu, mu, mu}
> > for all mu.
>
> Yes this is called the integrated likelihood. There is one mu, one
> sigma and many x_i(s). The x_i(s) are the data. In classical MLE
> estimation the mean(xbar)=sum(x_i)/n.
> and the sigma (standard deviation) is sqrt[(sum (x_i - xbar)^2)/
> (n-1)].
> What we are trying here is elimination of the nuisance parameter (mu)
> so that we can estimate sigma directly from the data (x_i). See
> Reference:
> Berger. Integrated Likelihood Methods for Eliminating Nuisance
> Parameters. Statistical Science 1999, Vol. 14, No. 1, 1-28.
First, you need to compute the definite integral, {\[Mu], -Infinity,
Infinity}. (Use GenerateConditions -> False or Assumptions -> \[Sigma]
> 0.) Second, I don't think Mathematica will compute the integral for
the symbolic product, so you will have to give a specific value to n.
Third, you will have to make the substitution of xbar yourself.
--Mark