# likelihood – Parameter estimation using LogLikelihood

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.