🚀 go-pugleaf

I am posting this for a collegue.  Please email me with your responses
and I will pass them along.  

Question follows:

I have a parameter estimation problem which sounds simple but 


The following question is being posted for a collegue.  Email me 
your responses and I will pass them along...

I have a parameter estimation problem which sounds simple but 
seems to have been missed in the literature.  We have a Gaussian 
random variable X with mean and variance given by the unknown 
parameters mu_x, and sigma_x.  We get observations Yi = X + Ni.
The Ni are independent, non-identical Gaussian random variables 
with varying but known variances.  Our observation Yi consists of 
the observation Yi and the variance of the particular noise at 
time i, what I call sigma_i.  I have derived the MLE equations 
for mu_x, and sigma_x and find that they are non-linear and not 
solvable in closed form.  While they can be solved for numerically 
we are working in real time and cannot afford a Newton-Raphson type 
numerical solution.  I have also worked the problem from a linear 
least squares approach, and while this works OK for estimating mu_x, 
they do not work for estimating sigma_x as sigma_x is an inherently 
non-linear function of the observations.  Can anyone direct me to 
any literature regarding this problem.

Thanks in advance for your help,
Mark Levin