In a scalar parameter estimation case, if the covariance of the observed data is involved with parameter, the distribution function (likelihood function) or the joint distribution of the observation becomes much more complicated. Set the derivative of it to zero and are we guaranteed an unique solution of the parameter? In my case, the derivative is the cubic function of the parameter and there would be three roots. Is it for sure that there would one real and two complex roots?
Dear Huiping Zhuang,
In general the soluition Is not unique if the derivative of a likelihood function in the form of the Gaussian distribution set to zero. But in your case, as I understand it, the solution is unique because there is only one real root as you pointed out.
Hi Sergiy,
the solution of a cubic equation can be of three different real roots and one real root. I don't see how that would be unique.
Bset regards
zhuang
Dear Huiping Zhuang,
If the solution of a cubic equation has three different real roots, it is necessary to investigate stationary points of (values of the roots). Find the second derivative of the likelihood function and choose those roots for which its value is less than zero. If there is more than one root, it is necessary to calculate the values of the likelihood function for these roots. It is necessary to choose the root, for which we have the maximum value of the likelihood function.
Dear Gergiy,
The diffculty is that the second derivative of the likelihood function is too comlicated to compute. So maybe just compute the numerical solution and substitute it into the function to seek the disired solution?
zhuang
Dear Zhuang,
You can calculate the value of the second derivative numerically.
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