[R] Estimating bivariate normal density with constrains

[Rolf Turner]

On 20/10/11 02:31, Serguei Kaniovski wrote:
>
> Dear R-Users
>
> I would like to estimate a constrained bivariate normal density, the
> constraint being that the means are of equal magnitude but of opposite
> signs. So I need to estimate four parameters:
>
> mu    (meanvector (mu,-mu))
> sigma_1 and sigma_2 (two sd deviations)
> rho (correlation coefficient)
>
> I have looked at several packages, including Gaussian mixture models in
> Mclust, but I am not sure what is the best way, or the best package to use
> for this task.
>
> Greatly appreciate any suggestions!

I very much doubt that there is anything built-in that you can use.
However it shouldn't be *too* hard to get maximum likelihood estimates
using optim() to maximise the (log) likelihood.

For starting values I would try just using the ordinary covariance 
matrix estimate to get your sigma_1, sigma_2, and rho estimates,
and for mu use (x1.bar - x2.bar)/2 (in what I hope is an obvious
notation) for a starting value.

Good luck!

	cheers,

		Rolf Turner