If I have a likelihood function with four parameters to maximise where two parameters comes from distribution function under study and other two parameters come from certain assumptions. Then is there any method to find best estimate or to optimise estimates, based upon some criteria like unbiasedness, consistency, efficiency, etc.
If you have fixed assumptions about the values of two of the parameters, then you have only 2 parameters to optimise. So I don't get the problem. Could you be a bit more specific: how does the model look like, what are the assumptioons? Are you looking for a numerical solution (if so: how?) or is your problem to get the partial derivatives of the likelihood function etc. ?
Ahlam H. Tolba and Jochen Wilhelm thank you for showing your interest in the question. I am studying lifetime models, to be more specific my interest is to estimate the parameters (2 parameters coming from the distribution; let say A and B) of competing risk model with two risks competing for the life of the individual/unit at the same time. In addition to this the lifetime is assumed to be a fuzzy number(remaining 2 parameters come from here; let say a and b). Now, the problem is that till now I have optimised the likelihood for parameters 'A and B' keeping 'a and b' fixed. But on changing values of a and b arbitrarily it has been observed that bias and mean squared error was changing. And bias was even changing it's nature from negative to positive. So my question is can we estimate a and b as well to get the unbiased estimates or more consistent estimates or more efficient estimates?
Why would you "change a and b arbitrarily" when these ought to be fixed by some theoretical assumption?
If there is no theoretical justification for the values of these parameters, why not taking the optimum likelihood in the 4-dimensional parameter space?
Actually when I run simulation I have a data generated in a certain format which has some incompleteness incorporated. This incompleteness has been treated using fuzzy numbers. So from the very beginning there is no true value for 'a and b' with which we can compare the estimates obtained in 4 dimensional parametric space. I have given it a thought that I should optimize likelihood in 4-dimensional parameter space but I am not very sure whether this can be done or not.
Hi,
I think you should use the MCMC 'Markov Chain Monte Carlo' where the fixed values are following the uniform distribution and the other two of the parameters assume to be exponential since it is a lifetime.
@Eman Alawamy thank you for showing interest in the problem but what you are suggesting is a Bayesian approach and I want to work in classical framework.
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