I’m trying to understand Likelihood methods with parameter estimation in mind. To this end I am trying to construct small examples that I can play with. Let’s say I have some data which I know (or suspect follows the function)
$ $ f(x) = (x + x_0)^{2}$ $
and I want to find out the value of the parameter $ x_{0}$ and the associated error using likelihood methods.

Let us then make some pretend experimental data:

f[x0_, x_] := (x + x0)^2
  
ExperimentData = Table[{x, f[-1.123, x] + RandomVariate[NormalDistribution[0, 0.25]]}, {x, 0, 3, 0.1}];

Then let us construct some test data where I “guess” my parameter $ x_{0}$ . I replace $ x_{0}$ with the parameter $ \theta$ to represent my test value:

TestData = 
Table[
        {\[Theta], Table[{x, f[\[Theta], x]}, {x, 0, 3, 0.1 }]},
        {\[Theta], 0.5, 1.6, 0.1}
     ];

How can I use LogLikelihood to make to make a parameter estimation of $ x_{0}$ . Using my TestData? The motivation is if I cannot construct a pure function, for example if I generate my test data from a numeric intergeneration.

My approach so far is to maximise the log-likelihood of the “residuals”

X = ExperimentData[[All, 2]];
MLLTest = 
  Table[
        \[Theta] = TestData[[i, 1]];        
        F = TestData[[i, 2]][[All, 2]];
        MLL = 
    FindMaximum[
      LogLikelihood[NormalDistribution[\[Mu], \[Sigma]], 
       X - F], {{\[Mu], 0}, {\[Sigma], 0.25}}][[1]];
        {\[Theta], MLL},
        {i , 1, Length[TestData]}
    ];

Then if I plot the Maximum Log-Likelihood as a function of my guess parameter $ \theta$ .

However this is clearly wrong, so I think I misunderstand something about the Log-Likeihood in this context.