Can we Maximum log likehood for finding parameters' equation? I want to use MLE on matlab to conduct to the equation for three variable m, sigma_0 and sigma_th how can I do it

Hi I wonder can we use MLE syntax on Matlab to conduce the parameter equation for equation (1)? If you guys want more information.

The eq(2) is the MLE for eq (1) and to conduct to equation of 3 paramters we need to have derive eq (2) and let them equal 0. That is what I know. If it can please help me at %solve MLE for probability of f(x).

In case, MLE can't do it. How can I simplify and eliminate all 3 derive equation, which I included at % simplifying and eliminating to find 3 parameters' equation.

clc; clear; close all;
syms m sigma_0 sigma_th sigma i N


%solve MLE for probability of f(x)
f=(m/sigma_0)*((sigma_i - sigma_th)/sigma_0)^(m-1)*exp(- ((sigma_i - sigma_th)/sigma_0)^m)

m'=MLE(f,m)
sigma_0'=MLE(f,sigma_0)
sigma_th'=MLE(f,sigma_th)
-------------------------------------------------------------------------------------------------------------
% implifying and eliminating to find 3 parameters' equation.  
% Define x_i
x_i = (sigma - sigma_th) / sigma_0;

% Define lnL with summation notation
lnL = symsum(log(m) - log(sigma_0) + (m-1)*log(x_i) - x_i^m, i, 1, N);

% Compute partial derivatives
dlnL_dm = diff(lnL, m);
dlnL_dsigma0 = diff(lnL, sigma_0);
dlnL_dsigmath = diff(lnL, sigma_th);

% Display results
disp('Partial derivative of lnL with respect to m:')
pretty(dlnL_dm)

disp('Partial derivative of lnL with respect to sigma_0:')
pretty(dlnL_dsigma0)

disp('Partial derivative of lnL with respect to sigma_th:')
pretty(dlnL_dsigmath)

[Torsten]

I gave you the solution here:

https://uk.mathworks.com/matlabcentral/answers/2175116-i-have-been-trying-to-derivate-equation-under-to-find-the-formula-for-3-parameters-weibull-distribu?s_tid=srchtitle

All you have to do is to substitute your data vector "sigma" for the symbolic array "sigma" in the code I provided and use "vpasolve" on the system

diff(lnL, m) == 0;
diff(lnL, sigma_0) == 0;
diff(lnL, sigma_th) == 0;

to solve for the three unknown parameters.

The system of the three equations is nonlinear - so you cannot expect an analytical solution. That's why you have to insert your data vector.

[Trong Nhan Tran]

Does it work as simultaneous equation method?

[Torsten]

It works as

dlnL_dm = subs(dlnL_dm,sigma,data_vector);
dlnL_dsigma0 = subs(dlnL_dsigma0,sigma,data_vector);
dlnL_dsigmath = subs(dlnL_dsigmath,sigma,data_vector);
[m,sigma_0,sigma_th]=vpasolve([dlnL_dm==0,dlnL_dsigma0==0,dlnL_dsigmath==0],[m sigma_0 sigma_th])

a system of 3 equations in 3 unknowns and Newton's method to solve.

data_vector is your data array of size 1xN.