I have data set consist of x and y variable where I wan to perform maximum likelihood estimation (MLE) to fit a sigmoid mean function μ(X;β)=β0​+1+e−(X−β2​)β1​​ and a linear standard deviation function σ(X;α)=α0​+α1​⋅X to the data. I am estimating the the beta and alpha parameters using Maximum likelihood function L(θ∣X,Y)=i=1∑N​logf(Yi​∣Xi​;θ) in order to observe the mean and sigma trend in my data.
likelihood = -np.sum(np.log(norm.pdf(Y, mu, sigma)))
And finally performing the optimization of the likelihood using result = minimize(likelihood, initial_params, args=(X,Y), method=’L-BFGS-B’,options={‘maxiter’: 100}) Every time I encounter the issue of obtaining different values for the mean and sigma when passing different initial values, I find that my model becomes sensitive to the initial parameters. Is there any way to address this situation so that I can obtain the best values for the mean and sigma that automatically fit into my model?
P.S. I have applied different optimisation methods also but didn’t work.
i have achieved required results by implementing it through other techniques but I want to implement it using MLE. How can I cope this issue ?

I have tried different optimisation methods