How to find closed form solution involving a sum over sample data

I am trying to find closed form solutions for various distributions with left-truncated data, i.e. I have to maximize the likelihood function

.

Finding a solution to this optimization problem numerically has proven difficult for me so I was hoping I might be able to just find a closed form solution. Is there a way in Matlab to just sum over some "n numbers", the value of which is in some range (i.e., > 0, < 50, etc.), and to then find a solution to some equation containing such a term?

Thank you for your help!

Answer 1

[Nipun]

Hi Jan,

I understand that you want to get the closed form solution of the given summation (the likelihood function) in MATLAB.

MATLAB does support closed form solutions to many ODEs and symbolic math equations. For closed form of "summation of polynomial series", refer to the MathWorks documentation below:

https://in.mathworks.com/help/symbolic/sym.symsum.html#:~:text=k%E2%89%A40-,Summation%20of%20Polynomial%20Series,-Input%20Arguments

However, in the given summation, the function "f(x)" is unknown. Therefore, it is not possible to get a closed form solution for this expression unless "f(x)" is explicitly stated.

Nonetheless, one can use the property of logarithms to compress the summation of logs to log of products. The final expression achieved is as under:

Hope this helps.

Regards,

Nipun