Question: LogLikelihood for a Bivariate Distribution

 

I have a Bivariate Distribution that has a PDF equal to 

 

(1/2)*exp(-RootOf(erf(_Z)*b*exp(exp((-x+a)/b))+b*exp(exp((-x+a)/b))-2*exp((-x+a)/b))^2-RootOf(erf(_Z)*sqrt(Pi)*d+sqrt(Pi)*d-sqrt(2)*exp(-(1/2)*(-y+c)^2/d^2))^2)/Pi

 

(1/2)*exp(-RootOf(erf(_Z)*b*exp(exp((-x+a)/b))+b*exp(exp((-x+a)/b))-2*exp((-x+a)/b))^2-RootOf(erf(_Z)*sqrt(Pi)*d+sqrt(Pi)*d-sqrt(2)*exp(-(1/2)*(-y+c)^2/d^2))^2)/Pi

 

How can I find the optimal parameter values of a,b,c,d with the LogLikelihood function for any given data set ?

 

I know how to do it for a Univariate Distribution :

 

restart;  with(Statistics); with(Optimization);

S := Sample(RandomVariable(Normal(4, 5)), 5000);

F := LogLikelihood(Normal(a, b), S);

evalf(NLPSolve(-F, a = 0 .. 5, b = 1 .. 6), 2)

 

but how do I do it for a Bivariate Distribution ...?

 

 

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