[R] Help finding the proper function

[Tom.O]

This might not be the correct forum for this question for there might be some
flaws in my logic so the R function I'm looking for might not be the
correct, but I know there’s a lot of smart people in this forum so please
correct me if I'm wrong. I have been googling and searching in this forum
for something useful but so far I'm out of luck.

This is the background to my problem. I have a set of samples that I know
are either from a normal distribution or a bivariate normal distribution and
my goal is to find the combination of samples that would give the best fit
of a bivariate distribution.  

I'm currently using maximum likelihood estimation to fit the bivariate
normal model but this is where my problems start. How do I find in an
efficient way the best combination, is there a function would do the trick
for me? 

One solution is to run an exhaustive search, but this would take a while
since the possible combinations is huge. So hopefully this is my last
option. 

My other problem is what test should I use to rank the models, WALD, F-test
or likelihood ratio-test (LR-test)? My colleague thought that the LR-test
would be the best to use, but he was not sure. And in that case which
function is best to use. I have found some LR tests but they use fits from
glm models etc.

Here is an example of my problem.
library(mixtools)

x.1<-rmvnorm(40, c(0, 0))
x.2<-rmvnorm(60, c(3, 4))
x.3<-rmvnorm(100, c(0, 0))
X.1<-cbind(rbind(x.1, x.2),x.3)
colnames(X.1) =LETTERS[1:4]

sample A and B is bivariate and C and D is not, so theoretically the best
combination would be to use A and B in the model since they change at the
same time, but other combinations with a bivariate and non bivariate
combinations would also work but should give a worse fit than A and B. And
the worst case would be to fit a bivariate distribution to C and D.

So this is the case...

Regards Tom

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