I have a highly non-linear (lots of jumps) likelihood function with K parameters (For example, a marked Hawkes Process used in seismology study). I implemented the L-BFGS-B optimization routine and it returns a Hessian matrix that has elements close to zero. As a result, several estimators of this likelihood function have large estimated standard errors.
Maybe the best way to describe the problem is that I have a Hessian that is nearly singular in the sense that its smallest eigenvalue is very small relative to its largest one.
I believe it is a optimization problem rather than a modeling problem and come up with an idea that in the first step, I fix some parameters with initial values, say a1,a2 and a3, and optimization others,say a4~a8. Once I got these estimators, I in turn fix them and estimating those who are fixed at first place (a1~a3). Iterating these two steps until all of them converge to a set of stable estimators.
My questions are 1) Is this idea legitimate ? i.e. Are estimators consistent ? 2) If it is legitimate, should I use 8*8 Hessian matrix to compute the variance-covariance matrix or I use 3*3 Hessian matrix to calculate the variance-covariance matrix for the first estimators and then use 5*5 Hessian matrix to calculate the variance-covariance matrix for the other estimators?